It would be terrible for everyone involved. Valve would die, EA would die.
Luckily Valve would never do it or want to do it. In their handbook they list one of the flaws of the management system is it only works with a very specific type of person whose got a very unusual skillset for the industry, and that even one or two people who don't fit into that system can cause them a huge amount of internal problems.
I think it's safe to say, EA with their hundreds and hundreds of employees will have many people who don't have that very specific unusual skillset.
And then think about it, who in Valve has experience in publishing games? In managing studios across multiple locations? Who would sort out their taxes? Do they have the financial wizards required to do this sort of thing? Do they know how to perform a shareholder meeting?
How can you remove the entire management structure of a huge multinational entity and then replace it with a company that has no management structure whatsoever and have it work out? And if Valve plays 'nice', they don't pump out yearly sequels, they don't run franchises into the ground, where are they going to find the finance to employ all those hundreds and hundreds of people to survive the first 3-10 years when companies readjust, create new games the slow way and find markets for those games?
It would be like putting the painter Monet in charge of making a nickelodeon cartoon. He doesn't have the right skillset to manage that sort of production line and you can't actually afford to have someone individually crafting masterpieces in every single frame. The things that make Monet good can't translate to something that requires manufacturing loads and loads to survive