What is the closest hotel to USH? Hilton or Sheraton? And are either of them walking distance to the gate?
Both the Hilton and Sheraton connect with each-other, by a set of elevators. Furthermore, you would likely have shuttle that can take you quickly to the park. That said, if you don't want to wait for Shuttle, there are two options.
The first, if you are walking from Sheraton, is to go to the elevator's connecting that and Hilton, then to walk to the entrance of the Hilton facing Universal Hollywood Drive. From there, there are two ways you can do.
One, is crossing over Universal Hollywood Drive at the 4 way right next to Frankenstein Parking/Valet, and the other is taking a pedestrian bridge next to an overflow lot of the Hilton (which is as a potential lot of expansion for Hilton). Crossing the 4 way is about .28 miles (1,487 feet to park), and the pedestrian bridge is the similar amount of miles, with only the feet off by about 4 feet off from each-other.
Once again, Shuttle is optional if you don't want to go through the efforts of walking from there to Universal and vice versa.
If you want the closest, it's the Hilton.