He wasn't killed instantly and was taken into the ER to be treated. When Cristina heard that he had shot Burke, she tried to resuscitate him despite the odds, not wanting him to get out of it by dying.

Does Doctor Burke die in GREY's anatomy?

Burke dies, Version 1

Cristina turns out to be his doctor, and she finds the old feelings she had for him return. Sure, he's on his deathbed, but this is true love.

What happened to Burke on GREY's anatomy?

Several episodes later, Burke got shot while trying to save a patient. The shooting brought the couple closer, as Cristina helped nurse her boyfriend back to health. RELATED: 'Grey's Anatomy': Fans Argue Over Which Exit Was More Depressing—Cristina Yang or Derek Shepherd? In season 3, Burke proposed to Cristina.