What is a public airport?
You can check out this info from the FAA

And the official FAA definition: 


  Public use airport is an airport available for use by the general
  public without a requirement for prior approval of the airport owner
  or operator.


It should be noted that 4 airports in the US are considered "High Density" and have special rules about operations during certain times which require approval for use (arrival/departure) despite being public. 


  §93.125   Arrival or departure reservation. 
  
  Except between 12 Midnight and 6 a.m. local time, no person may
  operate an aircraft to or from an airport designated as a high density
  traffic airport unless he has received, for that operation, an arrival
  or departure reservation from ATC.


In answer to the question as its now worded, 

No, there are public airports that are not federally funded. The airport I currently fly out of KDYL is owned and operated by the local government (Bucks County) and they are responsible for most things that go on there. I am not sure if they have received Federal funding in the past but I will ask around.

There is also such thing as "Privately Owned/Publicly Used" (elsewhere called POPS) airports like Trenton Robbonsville N87. These airports are owned privately, run privately, and generate their own income however you don't need to prior approval to land there and are open to general public use.