- I can't do anything rather than waiting.
- I can't do anything rather than wait.
Which one is correct and why?
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Idiomatically (and logically, it seems to me) neither of OP's suggestions are valid. It should be...
I can't do anything other than wait (or but/except wait)
Valid constructions using rather include, for example,...
I would/will do anything rather than wait.
?I'll do anything rather than waiting.
I've marked the second one there with ? because it doesn't sound very good to me, but I'm pretty sure at least some native speakers will find it perfectly acceptable. People have different ideas about where (the increasingly common) use of gerunds is "reasonable".
Neither is acceptable, at least in U.S. usage. Ordinarily, other than or but and not rather than is used in sentences like these. Also, I'd prefer “can do nothing other than” or “can do nothing but” to “can't do anything other than”.
After those substitutions, the phrasing with wait is closer to grammatical correctness. Consider the following conversation, and imagine it with waiting rather than wait.
A: What will you do?
B: I will wait because I can do nothing but wait.