I have a couple of comments on this:
1) You say you put him on Ax, which means he flat called the flop with ace high and checked behind on the turn on a blank. This is a very common play in limit, but it's much rarer in no limit. Most times people call flops in position with just ace high, it's not just to play ace high for showdown value. It's usually some combination of a value call and a float, and in either case, I think they'd be betting the turn after you show weakness just in case. Now this kind of logic may not prevail at very low limit games, and he very well might have played a weak ace high that way, but it's something that you should keep in mind when you move up to more aggressive games.
2) You say you think he had ace high, and you bet the river so he'd "go away". It doesn't sound like you trusted your read very much. Unless you think he's going to call your bet with ace high, it's more profitable to check/call than to bet. If he's always folding ace high, you only get that extra bet in when you're beat. If you check, he's going to bluff his ace high some of the time, and you get an extra bet when you're ahead. Now, maybe he'll fold some hands that are better than yours if you bet, but that's a different read entirely. I guess my point is that you can either play just based on your hand strength, or you can make confident reads and play based on those. If you're not at the point where you can make confident reads and make gutsy calls and set up your opponent to bluff, then you shouldn't be putting money in on the river with ace high.