Tate defended himself as the two players went to the ground before players and coaches from both teams separated the two, who have a history with each other.
"First off, I don't want [co-owners] John Mara, Steve Tisch or anybody involved in this team to have to deal with something like this after the game," Judge said Monday. "This isn't why we play the game. We have 60 minutes to beat the hell out of each other legally between the whistles. We don't need anything extra after.
"I talked to a number of our players, and obviously, we saw some video on [it] and guys that were involved, and I kind of had an eyewitness standpoint for myself after me and Sean [McVay, the Rams' head coach] had an exchange at midfield. I turned around, and it was kind of happening right there in front of me. So I got a quick glimpse of it right there.
"Look, all I have to say is the account I got from a number of our players -- there is a history obviously between them, and there was a punch thrown. Golden was defending himself. I was told he wasn't the one who threw the punch. And everybody involved was trying to break it up.
"I can say that both our players and the Rams' staff and players, from what I saw with my own eyes, were there trying to break it up. I didn't see anything going on in the pile. ... Guys were trying to break it up."
The NFL is reviewing the matter, but suspensions are not expected for either player, a source told ESPN. The Giants don't plan to discipline Tate, barring any new information.
"There are a number of things we still have to look through as a team right here," Judge said. "But based on the information I was given and what I saw with my own eyes and what our players gave me, it didn't sound like there was an immediate need for [discipline for Tate]. But we'll look into it if there is. Obviously, we'll always take care of our players."
McVay said Monday that the sight of Tate and Ramsey trading punches after the host Rams' 17-9 victory caught him off-guard.
"You look over your shoulder, and it's melee," McVay said. "And those are things we want to avoid."
McVay said he spoke with Ramsey about the cornerback's role in the fight to make sure "we're all on the same page of we can't allow some of those things to get in the way of whether it ends up being something bad happened for you or for our football team."
"Jalen's a smart guy. There's a lot emotions," McVay said. "As far as the specifics, these are grown men. I didn't get into the 'he said, she said.' I just really was more along the lines of, let's be smart, let's make sure we don't let this take away from what we just were able to accomplish as a team."
McVay did not clarify if Ramsey faced any team discipline.
"Any of that stuff, we handle internally," he said. "But there's so many things that ended up occurring. I think it's more about, let's understand it, and let's make sure that we learn from this situation."
ESPN's Lindsey Thiry contributed to this report.