<p>Is there a way to find out what gcc flags a particular binary was compiled with?</p>

I don't think so.
You can see if it has debug symbols, which means -g was used ;) But I can't think of any way how you could find out which directories have been used to search for include headers for example.
Maybe a better answer is possible if you only target for a specific flag; e.g. if you only want to know if the flag "..." was set when this binary was compiled or not. In that case, what flag would this be?