Which of the branches of the armed forces in the U.S. is the most important today?

There is no room for fat in the services, all have an essential role. The fat is the procurement of equipment to do the work. We are working on keeping cost down but it is out of control the private contractor is fueling much of the pork. 60 Minutes had a segment on the doubling cost of fighter jets. They have problems and when they do we were paying to fix them. It is the way of most procurement, if the government needs it business bids high and until recent past have had issues.

The value of each branch being equal we can not cut even a little bit. The Marines are the the first response The navy gets them close and supports the effort. The army mopps up and maintains the ground we get. The air force air dominance and surveillance pinpoints targets. The coast guard first and last line of ocean defense plus the save lives of people that have problems. They track and monitor coastline traffic for criminals.

We need them all and depending on the situation one will always stand out.

There is no argument: the Navy is the backbone of the U.S. military.

The U.S. military at present is primarily a power projection force. It is not a defense force: it is an offensive force. Sometimes, that does include peacekeeping or deterrent actions for allies and some countries; a lot of the time, that is about overthrowing regimes, maintaining U.S. economic and corporate interests, etc.

In any instance, the fact that the U.S. has the leading global Navy, as Britain (the previous hegemon) had before it, is the basis of its military power. The U.S. is able to mobilize ground and air troops anywhere in the world with aircraft carriers. Submarines can carry cruise missiles for pinpoint bombing of targets. And nuclear submarines carry the most important aspect of the nuclear deterrent that the U.S. has and the biggest part of its second strike capability.

Without the Navy, the Air Force would not be able to project power globally, and the Army and Marines would be largely defensive. Small units can be dispatched by airplane and routinely are, but the only way to move large amounts of men and materiel is by sea.

One can argue that, insofar as the U.S. military is about subsidizing research and development and providing military contracts, the Navy isn't as important, but even there the Navy not only requires new technological developments but also demands large inputs of steel, large shipyards, etc. That contributes massively to U.S. industry and infrastructure.

It seems like most technology comes out of the Air Force- GPS, cooperation with NASA etc.