I don't think having a carrier is completely necessary. It would take the "relief carrier group" days, if not weeks to reach the stricken area, in addition it would be difficult to restock once it arrives at its destination, although it would make a very convenient floating helo-pad/command and control center. It would be very cool, but if we are looking to make the most out of a budget, they could be skipped.
How about a dedicated base with a runway. The US contribution could be to turn over an old USAF base. Plattsburgh AB would fit the bill nicely in my opinion, plenty of land, and access to the Atlantic by the Erie Canal and St. Laurence Seaway to boot!
On this base you would maintain four or five "regiments" of relief workers, divided between doctors, nurses, military engineers, infantrymen, police (former MPs if possible and of course a few chaplains from the world's major religions. Officers with experience in procurement and logistics would also be vital. You might even want to throw in a few lawyers if the relief force is going to be operating in a more civilized part of the world, as outside the Third World work tends to get bogged down in paperwork instead of mud.
Each regiment should be equipped with all terrain vehicles, satellite-linked communications, and the equipment to set up a field hospital within 48 hours of arrival. That is going to involve tents, cots, and of course portable power generators. A solar panel or two would not be bad for a photo op, but for real electrical generation diesel generators cannot be surpassed.
As for getting to the disaster area, I really do not think that you could do much better than a dozen C-130s, or even C-47s. They can carry a good payload, and have the capacity to transport disassembled vehicles and even helicopters. Furthermore, they can land on the most primitive runways likely to be encountered. It does not hurt either than they rather simple machines compared to more modern aircraft, and could be as easily serviced overseas as at home.
The transport aircraft could move our relief force within a reasonable distance of probably 5/6 the inhabited areas on earth. For the truly inaccessible a complement of two dozen Chinooks. They have great lift capability, and can fit disassembled into a C-130, and probably a C-47 as well. After being unloaded at the nearest airport, the Chinooks could transport the relief regiments to their destination, as well as the necessary supplies, most likely by shipping crates.
Bulldozers, tractors and trucks would probably be best acquired near the scene of the disaster. It would be absurdly difficult to airlift them, the cost in fuel would probably just about equal the cost of renting them on the ground.
Did I miss anything?