Well, for starters, nothing at all would happen for at least a year.
I think Valve would treat EA as their publishing arm, someone to fund third party developers' projects in exchange for a cut of the earnings. Most of the staff probably wouldn't change, except for the people at the top whose business practices run counter to Valve's philosophy. Most existing IPs would stay around. In short, they would remain separate companies for the most part, but EA wouldn't be quite as evil as they have been of late.
Eventually Valve will start trying to consolidate things, folding Origin into Steam and such. The creators of the Frostbyte engine will be folded into the development team for Source 2. Valve's push into the living room will mean that EA's console games will migrate to the "Steam Box", meaning future EA games will all have native Linux ports (and probably Mac as well). Sports franchises will become single games with annual automatic updates, and complaints will start pouring in from people who didn't want roster updates pushed on them and didn't realize you have to turn updates off ahead of time. And the
Battlefield franchise may be shelved due to concerns that it's competing with
Counter-Strike.