I'm no scientist, but the volume of water in the Med. is significantly less than in, say, Antarctica (total Mediterranean Basin, including land-areas such as Greece, Italy, et al, is 2,085,292 km², while Antarctica is 13,200,000 km², and that's ignoring Arctic ice deposits), so the Med. being empty wouldn't be the equivilent of the poles melting down or anything.
Not sure about the atmospheric effects - probably be one hell of a lot of thermals. That'd have some effect on global wind patterns for sure.
I couldn't find a lot more through some quick Googling, and these are all basically theories trying to prove or disprove the Mediterranean flood was Noah's Flood (or, at least mention it), but they also offer some decent information on what the basin itself would have been like, assuming this did in fact occur.
See here,
yo.
The notion of over a mile of salt coating the bottom of a boiling desert recessed deep below the surface of the earth is, frankly, kinda freaky.