It's applied to other civilizations pretty regularly.

The most cut-and-dried one off the top of my head is Sisko citing the PD when declining to help Tosk in "Captive Pursuit".

The Prime Directive and the rules governing first contact overlap, but they are distinct.

There have been plenty of indications that the Prime Directive applies to warp capable species. I think episodes like "Too Short a Season" and "30 Days" could be cited as evidence, though some would argue we're never explicitly told that either of the civilizations being interacted with are explicitly warp capable. In "Redemption" Worf resigns his commission after Picard claims the Federation cannot support Gowron in the Klingon Civil War, citing the Federation's, *"principle of non-interference." Granted, he does not explicitly say it's the prime directive. However, there is "The Outcast" the J'naii that Riker falls in love with, Soren, claims to be familiar with all the systems aboard a Starfleet shuttle, including the warp nacelles, and Picard later tells Riker he can't interfere with the J'naii subjecting Soren to conversion therapy because of the Prime Directive.

And, if you want the most explicit example, in the PRO episode, "First Con-Tact", a screen displays text -- copy and pasted from the book "Star Trek: Federation - The First 150 Years" -- outlining the general rules for how the Prime Directive applies to warp capable cultures.

"Section 2:
If said species has achieved the commensurate level of technological and/or societal development as described in Appendix 1, or has been exposed to the concepts listed in section 1, no Starfleet crew person will engage with said society or species without first gathering extensive information on the specific traditions, laws, and culture of that species civilization. Then Starfleet crew will obey the following.
 
a) If engaged with diplomatic relations with said culture, will stay within the confines of said culture's restrictions.
 
b) No interference with the social development of said planet."