I have had answers, such as this one, where the question asker comments below it and says something like: "Thanks, that did the trick!" but doesn't accept the answer. Is there anyway to flag a question as "apparently answered" so that the questioner can get a stronger suggestion to flag it as answered?
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There's no such flag.
If the user is very new and/or has a 0% accept rate, it might be appropriate to very gently suggest that they accept the answer that helped them solve the problem.
Rude comments are always unacceptable, and virtually any accept rate other than 0% (suggesting that they simply don't know how accepting answers works) is not okay to bother someone about.
Realize that users might not accept an answer for perfectly valid reasons (they're waiting to see if an even better answer appears) or even totally idiosyncratic reasons (they only accept answers with perfect grammar) and sometimes there's simply nothing to be done. Take quiet pride in having helped someone with a problem they faced and hope that the next person is more reciprocating.
This does make the unanswered questions list less useful as some of those questions actually are answered, but the feeling so far has been that potential fixes (like an "apparently answered" flag) will likely introduce more problems than they'd solve.