The CIA and military do drone strikes. One of them requires presidential approval, the other does not. I am not 100% sure on this, but I think the military strikes can be done independent of presidential approval.
Military strikes
within an approved theatre would not need independent approval, so long as they were including in the original approval. A military force in Afghanistan assigned a drone task force would be able to deploy it as they like. However, if the military were making a strike in Pakistan, outside any current operational arena, they would need specific permission. There isn't much difference between a drone strike and a special operations strike like the one that took out Osama from that perspective. The violation of Pakistani sovereignty is always going to need high level go-ahead.
I would note that the official stance, despite the leaks and recent press stories, is that, "whether or not the CIA was involved in drone strike operations ... is a classified fact." [Context -
An ACLU FOIA request.] Unofficial releases aren't seen as admissions of CIA involvement, even if they are extensive discussions of the legal and logistical structures of such strikes. That slightly complicates matters here, as it's almost certain this was a CIA lead strike but the administration are probably going to be vague on that front and treat it as a military operation. It would be (legally, politically) interesting if they came out and officially announced this was a CIA strike, but it seems unlikely to me at this time.
Unrelated;
Obama seems to be fighting clean.• Democratic presidential advertisers aired 35,936 ads. Of these, 70 percent (25,092) were positive and 30 percent (10,844) were negative.
• Republican presidential advertisers aired 27,857 ads. Of these, 27 percent (7,584) were positive and 73 percent (20,273) were negative.
So there's that.