First off, the doctor must wash his/her hands after entering the room, period. The doctor opened the door with his hand, right? That's a (major) point of contagion right there. Unless you're in some futuristic practice with automatic doors, I would never excuse the doctor from not washing post-entering the room.
Second, unfortunately, doctor handwashing is a major problem. Major studies have repeatedly found very low wash rates (This Study found a 40%-50% rate, for example), and the problem is very difficult to address from a management perspective.
From a parent's perspective, the odds are you can't really effect change here. That's not to say you shouldn't try; but I wouldn't presume it's likely to work. Doctors are creatures of habit, and if they're not in the habit of hand washing, they probably won't easily become in that habit - especially if only a few patients mention it.
What can you do? First, you should certainly mention it to the doctor the next time he comes in. "Hi, would you mind washing your hands before you examine my child?" Polite but matter of fact seems reasonable here. It's a reasonable request, given it's your child's health, and as long as you're not critical or condescending about it, the doctor shouldn't take it wrong. If that goes reasonably well (ie, the doctor doesn't object and does wash his hands), then continue that - each time. It may simply be that the doctor isn't in the habit and appreciates the reminder. If the doctor does take it badly, you may need to choose between a doctor with good hygiene and that doctor.
You also could wash your hands after the doctor comes in. That might serve as a subtle reminder, after all. If your child/children are old enough, you might encourage them to take an interest in this; doctors (particularly pediatricians) often are familiar with children who are a bit pushy about things around them (ie, most children) and could take it better from the child. This particularly works if you have a child like mine who enjoys washing and hates being dirty.
Finally, I would not address this with the practice administration or anyone other than the doctor. While it's possible some volume of complaints might initiate a change this way, it's more likely to irritate the doctor, and doesn't seem likely to be successful if the more direct method wasn't.