What can I do?
You should smooth things over with the boss if necessary and maybe set some ground rules for future interactions: "Boss, I'm sorry if I upset you about the logo. I totally get that it's your decision, and if you say this is the logo, then I'm on board. I'd like to always be able to give you my honest opinion about these things, and I figured that's why you hired me, but at the end of the day it's up to you."
Should I persuade him or just let it be?
From your description, this boss seems unlikely to be persuaded on this issue, at least right now, especially by you. Persuading him would mean making such a strong case that he comes around to see things your way, and the chances of that seem slim at best. Even if you could lay out your argument so convincingly that he'd have no choice but to agree, you might be forcing him to do something that he doesn't want to do, and that might lead to long term resentment. For that reason, I think it's best to let it go.
The logo is not going to look bad on your résumé — nobody will blame you for a logo that you didn't design or have the power to change. Maybe you'll lose clients, maybe you won't, but that's really the boss's problem to worry about.
Keep doing the best work you can. If the boss overrules you on every decision, then it might be time to look for work elsewhere, but if you like the job otherwise, just work around this roadblock.