The NATO campaign in Libya was started by the UK and France, and the US was later also involved. How significant and vital was the US assistance in this campaign?
Edit: I am primarily interested in the actual military role, not political, diplomatic, etc.
You are interested in the largely US Operation Odyssey Dawn. This contains the disposition of US forces and the role of United States Africa Command (AFRICOM). It also contains a summary of action taken by US forces, from bombing and ship-to-shore engagements on the 19th March to naval engagements on the 29th March, including a marine ground incursion to rescue downed pilots.
NATO Operation Unified Protector took over on 23rd March and lasted until 31rd October, 11 days after the capture of Sirte. Obviously NATO has large American element, but this change in operations came with the US intention to scale down their involvement significantly. I can't find a sumamry of action for this one unfortunately.
Here is the contributions and expense summary:
"USA: from 1 April to 22 August, the US flew 5,316 sorties over Libya, including 1,210 strike sorties, with munitions deployed 262 times. By 31 July, the US had spent US$896 million in the conflict." Citations on the wikipedia page
US forces commited on wikipedia as well.
It can be safely said that the US role was crucial. This is because it is the US who determines the foreign policy of NATO and its members in most important respects.
It is hardly possible that a US-affiliated country begins bombing or invasion in other countries without consent and support from the US.
The US was leading the diplomatic pressure on Libya. It was also behind the notorious UN resolution. It put pressure on Russia, China and Libyan neighbors so that they ceased support for Libya.
It was the US who organized the initial anti-government riots in Egypt and Libya through US-supported and instructed political activists. Note that the "Arab spring" protests starting from Egypt even used the flags with international logo of the "color revolutions" - the black fist on the white background, which has been featured in the French film "Revolution.com - USA: The Conquest of the East".
The USA was also among the first governments to recognize the rebel government.
Regarding the US share in the NATO-conducted military operation and military defeat of the government forces, the US contribution was also high. The planes which bombed Libya were launched from the US carriers, US provided infrastructure, coordination, nintelligence, planning and communications for NATO.
Besides this, it closed all satellite communications channels to the Libyan government and shipped weapons to the rebels.
