The union knew they couldn't hold the fort. Why didn't they just abandon it, or destroy it?
partly because of the shenanigans of then Sec. of War Floyd. When Robert Anderson took over Fort Sumter, he fully expected an attack by SC militia. But his messages to Floyd were pretty much ignored, and Floyd didn't even pass them on to Winfield Scott. After Anderson abandoned the Federal outposts on the mainland and moved all the Regulars under his command to Sumter, he recieved conflicting orders from various people in Washington... some saying he should surrender if attacked, some saying he should fight, and some saying he would be receiving reinforcements. Obviously, the people in DC were unclear of what to do.
The move to Sumter was actually rather admired in the north (and naturally demonized in the south)... the sentiment in the north clearly indicated that they thought Anderson was a hero and a patriot. Floyd was eventually forced to resign because of financial misdealings, and the President's Cabinet became dominated by Unionists... they eventually persuaded Buchanan to keep Sumter occupied and supplied. A relief expedition sent to the fort was fired on by SC militia, and turned away (these were arguably the first shots of the ACW). Basically, Sumter was kept by first Buchanan's vacillation, then the fact that Unionists took over the Cabinet.
When Lincoln took office, he took a stance that the secessions and Confederacy were illegal and states in rebellion, and thus, no surrender of Federal property would be allowed (in spite of that, CSA forces were seizing Federal territory all over the south); nothing that would give the CSA an air of legitimacy was allowed.
Probably what tipped the south into attacking Fort Sumter more than anything was a miscommunication from Seward to the south, where he practically promised that the fort would be evacuated. When the south found out that not only would it not be evacuated, but was going to be reinforced, the south felt that the north had broken a promise (although Seward was not officially speaking for Lincoln). Both sides were determined to avoid firing the first shot. One more reason the south did just that was because of worry (ah, irony) that SC would secede from the CSA if they didn't do something about Fort Sumter....