I know that ALL airports that have received federal funds are considered to be open to the public, however does that mean that every "public" airport was federally funded?
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.