Not sure which mission it is but it's on the deck of a ship with a giant spider boss. The game crashes and Windows informs me that I'm running low on memory... despite my having 6 gigs. Now there's no way Crysis is utilizing that much is there? What can I do to remedy this?

Crysis is/was a major resource hog and the 64 bit was even worse. If you have the latest drivers installed for your sound & video card, the only other problem I can see is that your HD TV can't handle the game after a while or you are running to many processes in the background while playing.

Let the game dictate the optimal settings for your rig.

make sure your pagefile is set to system managed and try running the game with superfetch disabled (this was an old fix for the same issue in BF2)