If you mean winning black's e5 pawn in the Ruy Lopez by the exchange variation, it's met with the double attack by queen to d5.
The battle for a pawn
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No, I know the ruy lopez line. What I'm talking about are the endless flurries of moves which are best described as "attacking pawn... defending pawn... attacking pawn... defending pawn... pinning defender... unpinning defender..." etc.)
Personally I don't like the pawn war, so I am not very familiar with it. I play openings that are more open and don't result in it.
I've been playing d4 as white a lot lately, but a lot of my opponents always open up with e4. I rarely defend against d4 :(
My usual defence against e4 is the French, Pirc or Sicilian but I've been enjoying meeting e4 with e5 again getting into the Italian or Spanish, only so I can play h3 to gain some luft for the king, prevent a pin, stop the knight coming in for a Fried Liver Attack (It's a multi-purpose move I think they call it).
There are times sacrificing a pawn is advantageous, others when you should hold equality. The fun part for us lowly amateurs is trying to figure out when we should gambit the pawn😀. That comes down to calculating well enough to know for sure the gambit will be beneficial by either a strong attack on the king or a large initiative that will be hard for your opponent to counter. Keep in mind sacrificing material usually means you don't want to reach a endgame, if your initiative fails to mate and opponent is a strong endgame player, you will probably lose.
I enjoy playing the queen's gambit but more often than not, at my rating, they always accept it! Not that it bother's me but it pisses me off because I want to play a queen's gambit declined line.
If they accept play e3 and dare them to defend the pawn, b5 a4 c6 axb5 cxb5 Qf3 and black loses a piece. It's a well known trap but just throwing it out there anyway in case you're unaware of it.
Usually, in these "pawns wars" that start especially with very classical lines like the ruy lopez, the person who wins the pawn gains initiative, attack, and activity as well. This is just a pattern in chess. It's my opinion the pawn is as unimportant as you say it is, the key thing is to get initiative and a good attack on the opponent's king. If you can do this by shuffling pieces around to defend your pawn, then do so. If you can achieve this by sacrificing the pawn, then sac it. As long as you have the correct objective in mind the right answer won't be nearly as mysterious.