Very roughly, FAA (not ATC) 24 aircraft per body, Australia about 8.5, based on BITRE estimate of active aircraft for 2015, and the FAA figure since the fleet wide rego/re-rego check, which eliminated most "hangar queens".
This isn't an easy comparison. While (based on the FAA Annual Report) you can identify the number of controllers, its not so easy to identify the equivalent roles to AsA in management, Navaid maintenance, etc. Also, The FAA employee numbers include many job functions that we have in the ATSB. It also has a certification function which is largely absent here. And in Australia, AMSA provides some of the services provided by the FAA in the US).
The 2014 report
http://www.faa.gov/about/plans_repor...14-FAA-PAR.pdf lists: (CASA numbers from 2014 Annual report after /)
Total Employees: 45,543 / 871 + 4475 + 30 (ATSB) = 5376
No; of pilots: 600,000 / 36,158
No. of Airports: 540 / 191
No. of aircraft: 202,000 / 15,529
In aggregate the ratio comparisons are (USA / Aus)
aircraft per employee: 4.43 / 2.9
pilots per employee 13.2 / 6.7
So, we don't look so good.