Everyone learns in their own way. I love going to classes/seminars/workshops, but for a while I dated a guy who couldn't understand why I would do that, as he learned best on his own, reading through material and studying it.
Your son is old enough now that his statement that he likes to learn things on his own reflects some good self-knowledge, and I would hesitate to undercut that. By Grade Four, he is also old enough to begin organizing things for himself, which will become a vital skill when he gets to middle school.
And, the fact is that he may well be able to learn both music and dance on his own. However, he will not be able to learn either one without plenty of practice. So, maybe you can compromise with him, and let him set up his own "curriculum" for learning these skills over the summer. Let him pick out a goal that he wants to accomplish--maybe it is playing a particular song that he really likes on a musical instrument of his choice, or maybe it is creating a dance routine to a favorite song. Go hunting on YouTube for videos of people who are doing the activity that he wants to learn--and make him a playlist of videos that he can use for inspiration. Help him break down the steps he will need to take to achieve his goal (because that is the organizational skill that Grade Fours are only just beginning to develop). Set up a type of worksheet to show what he needs to accomplish every week so that he can get to his goal. And then let him organize his time as he wishes--but when he misses meeting a weekly goal (as he is very likely to do), sit down with him and talk about how he could have better organized his time (less TV watching? Less video game playing? Less...?) to achieve his goal. Don't get angry, just help him work on breaking things down into smaller chunks of time that are easier to achieve.
When my kids were small, I used to buy them Lego kits. At that time (the early 2000s) there didn't seem to be any general Lego building sets--everything was a kit designed to build something specific, like a castle, or a spaceship. And the kit came with a set of pictorial instructions to assemble the item correctly. Frequently my kids would have difficulty following the instructions and I would have to help them figure out how to put a particular item together. This was so different than the experience that I remembered with Legos as a kid, where there were just a bunch of blocks, with windows, doors, wheels, etc. and you were left to make whatever you wanted with the building blocks.
I bring this up, because I feel as if this is also something that permeates the lives of children in other ways, and seems to be underlying your question. I think there is a real tendency to overschedule/overstructure children's time nowadays. They have multiple after school activities, and during the summer they go from one camp to another. With my kids and their friends, I found that this overscheduling really impeded their ability to be creative and self-sufficient.
My summers as a kid were all about finding activities that kept myself entertained--getting together with friends, playing games, reading books, running around in the sprinklers, etc. And by the end of the summer, I was tired of coming up with activities for myself, and was eager to get back to school. My kids didn't have much of an opportunity to figure out how to entertain themselves--and I think that did them a disservice.
That is not to say that you shouldn't expect your child to learn something over the summer, if that is something that you think is important. But, I think it is also important to allow them some unscheduled time to sit and think, to explore things that they want to explore with no expectations, and to get bored.