You have a wonderful set of conflicting goals here that I'm sure many parents have grappled with - I know I have.
I think your question really has three parts to it (so my answer is quite long - sorry, but I really hope it helps), the most obvious question is, should I make him apologize? but there are two other key ingredients here too: Will my coldness when he doesn't apologize be harmful to him ultimately? and How do I get the tantrums to stop without being cold with or without a forced apology?
I'll start with the coldness part of things. You are absolutley right, being cold teaches your son that your love is conditional. This is not a healthy thing for him at all, nor is it teaching him a better way to express his frustrations. You may find this article from the New York Times interesting, helpful and informative as it summarizes what the consequences for teaching children conditional love are likely to be.
Quote from the article:
It turned out that children who received conditional approval were
indeed somewhat more likely to act as the parent wanted. But
compliance came at a steep price. First, these children tended to
resent and dislike their parents. Second, they were apt to say that
the way they acted was often due more to a “strong internal pressure”
than to “a real sense of choice.” Moreover, their happiness after
succeeding at something was usually short-lived, and they often felt
guilty or ashamed. . . .
In practice, according to an impressive collection of data by Dr. Deci
and others, unconditional acceptance by parents as well as teachers
should be accompanied by “autonomy support”: explaining reasons for
requests, maximizing opportunities for the child to participate in
making decisions, being encouraging without manipulating, and actively
imagining how things look from the child’s point of view.
The last of these features is important with respect to unconditional
parenting itself. Most of us would protest that of course we love our
children without any strings attached. But what counts is how things
look from the perspective of the children — whether they feel just as
loved when they mess up or fall short.
The jist, "don't use your affections as a reward, or your removal of them as a punishment."
In regard to the other two questions
I'd suggest "starting over." for a little while. Often, when kids express themselves through tantrums it is because they haven't learned an alternative way to express themselves constructively. Although a tantrum isn't getting your son what he wants, it does allow him to be certain you know he is unahppy with you and the limits or decisions you made. Therefore, it is working for him on some level.
Surpise him next time he starts in on a tantrum and give him a hug along with an alternative way to express himself. "I know you feel angry right now, I love you." along with a hug, is likely to shock him back out of his tantrum given the history of things and will signal something is changing. You can then talk to him about "constructive" ways to express what he wants vs. inappropriate ways and how one might lead to win-wins while the other does not. Of course this also means he needs to understand the idea behind compromise is that everyone gets something but not necessarily exactly what was initially wanted. Then, state what it is he is wanting and that he is feeling frusted back to him. In doing this, you do two things: Let him know you heard and understand him and still love him even though you disagree, and you model constructive ways to express his emotions.
After this first experience, don't give him a hug every time, but if he starts in on a tantrum you might try, "I know you are frustrated that you can't have (said thing)." When used in combo with the "win-win" approach I am about to describe, you will teach your son the alternative way of communicating he needs.
You can do modeling seeking out win-win solutions yourself with he, any siblings and with your significant other (if present), but you can also further your child's understanding through ongoing conversation and/or stories in which compromises are contained.
The frame to use with him - verbatum at first so he starts hearing it as a pattern is: "I understand that you want (said thing), what I want (know you need) is (thing he needs), I'd like to try (a workable solution that also aknowledgges what he wants." Start out offering up a solution that you are happy with and aknowledges his wishes too.
A real-world example of this might be: "I know you are in the middle of a game right now, but dinner is ready for us to eat. How about I give you two minutes to get to a good stopping place in your game and then we'll eat? I can do that." You've let him know you've heard his wishes. You've stated yours. You've offered up something that shows you are giving a little if he gives a little (do not do this in response to tantrums though - at first, you'll need to do it pre-tantrum so it requires a little prophesy on your part).
When you've done that for awhile give him the chance to come up with the mutually agreeable solution with "I understand that you want (said thing), but I want (know you need) for (what you want/think he needs), so can you think of something that meets both our goals?"
An example of a real life situation: "Son, I know you want a snack right now, but dinner is only 30 minutes away. I need you to have a healthy meal and not be too full of snacks for dinner time. Can you think of a solution that meets both our wishes/needs?
of course there are times when as the parent, there is no compromise because you are concerned about safety, health or values you hold dear and are trying to teach but if you compromise on some of the other things, you can honestly say you do, and this is just one of those times you have to pull the parent card because, "I love you son and know and want what is best for you."
A related set of resources which my husband and I refer to frequently are the Seven Habits books. In particular, we use "The Seven Habits of Highly Effective Families" and "The Seven Habits of Happy Kids."
In regard to apologies
Now that my own daughter is seven, here is how I've generally approached it with her.
She understands that I feel, when a person makes a mistake that causes some one else a problem or hurt, we have to do our best to make up for it or some how make it right. Often, the first step in doing this, is communicating to that other person who was affected that you would like to make it right. This can come out as an, "I'm sorry" but there are other ways to communicate the feeling too. "I wish I had not acted that way," or "I regret my actions made you feel. . . " are two others.
She also understands through conversations about it, my modeling and enforced consequences, that the apology is only the first step. An apology without congruent follow-up actions becomes meaningless, just like promises broken multiple times makes future promises meaningless. It is up to her in a given situation to figure out what the follow up action would be. As an example, she once dropped my camera and it broke. It was completely an accident, but she offered to buy me a new one - that is a congruent follow up action. Of course I thanked her for her offer and didn't take any money from her since it was purely an accident but the thought was there.
Dealing with the Tantrums Themselves if they continue after introducing all of the above
If you have aknowledged his wishes and then stated yours and he throws a tantrum in response, I would respond with, "I love you, but I do not discuss win-win solutions with someone that speaks to me that way. Would you like to try again?" At that point he may very well surprise you with a completely reasonable and rational win-win suggestion. If not, then, don't bend. Calmly and simply interupt anything he says with a raised voice or that contains an insult with, "That is not going to get you anything." Again, don't bend, he doesn't get what he wants, no win-win is discussed further and if he really gets out of control, he needs to go calm down in his room until he is ready to be reasonable.
Honestly, I wouldn't require the apology for coming out. I'd give him a hug or some loving remark about being glad he decided to rejoin you. Be glad to see him now that he is calm. At the same time, whatever it is he misses out on while he was calming down - he misses out on. If he is in his room while you are eating and misses dinner, he goes hungry for the evening. Don't pause favorite shows or games, just calmly move on with life. When he does come back out, you can say, "I'm so glad you've rejoined us, we are. . . " The key is that you cannot get emotional about it. When he finds you've turned off the game, or allowed the show he was watching to play through or that he has missed dinner or whatever, "I know, it is so sad your choices meant you had to miss out on _, maybe next time you'll find a constructive way to express your frustration." If you respond with emotion it is working for him.
If the tantrums don't stop after you've set the "refresh button" and started again with the focus on searching for "win-win" solutions, I'd take it to the next level with him in terms of lost activities. In adult life as well as his friendships, if he were to throw a tantrum every time he didn't get his way, people would stop inviting him places so:
Has he thrown any of these tantrums when it was time to leave a friend's? At the park? etc. etc.? Start not going to those places. "I'm so sorry son, I wish I could take you, but when you act unreasonably and throw a tantrum it makes me feel embarassed and I can't trust you not to do that. As sad as we all feel for you that you can't go out and enjoy (said thing), we have to learn we can trust you first." Here is where he starts figuring out there are real-life consequences for behavior that is out of hand, but you are being loving and warm about it, just matter - of fact and not saving him from his own actions.
Same thing at home. Is there a pattern to the tantrums? Is it usually over not getting to play a certain game? turning the TV off at bed time or meal time? sharing appropriately? "I know it is really a bummer that you can no longer watch TV after school, but you've shown us you aren't ready to handle turning it off when it is time to do homework, have dinner and get ready for bed, so, until we can trust you to make better decisions, we just aren't going to have it on at all. I know, its really a bummer we can't trust you not to have tantrums about it, but there it is."
Then, allow him to put two and two together and offer up the apology completely of his own volition. When he does, reward him with a little renewed trust and have a talk about how he can win back the rest of your trust. Good Luck and let us know how it goes.