Does a military equivalent to civil ATC exist?
Yes, for their airspace.

Generally speaking, US Military has its own ATC for domestic military installations and other venues such as US Navy aircraft carriers. Most of these military-staffed airports are rarely transited by civilian aircraft, and are instead designed to let the military run their own show with their own aircraft.

At some airports around the country which provide provide joint military-civilian use, such as Charleston Air Force Base in South Carolina, Scott Air Force Base in Illinois, and Marine Corps Air Station Yuma in Arizona. These airports have significant military traffic and also allow civilian aviation operations. At such airports, typically FAA controllers are in place and military aircraft work with them.

Most of the airspace in the US is owned by Air Route Traffic Control Centers, which are staffed exclusively by FAA-employed civilian controllers. These controllers handle all air traffic between airports, including military aircraft transiting the national airspace system, which means that civilian and military facilities all around the country must interact with each other on a daily basis. As a result, both civilian and military air traffic controllers use a commonly shared phraseology.

For exclusive military use, there is special use airspace which includes:


  ... restricted airspace, prohibited airspace, military operations
  areas (MOA), warning areas, alert areas, temporary flight restriction
  (TFR), national security areas, and controlled firing areas, typically
  up to FL180 or 18,000 ft above sea level.



Source

As discussed in this question, supersonic operations in the US are also performed in airspace which belongs exclusively to military.