Boyce and McArthur had to be separated by captain Gary Caldwell during the 4-0 home defeat by Liverpool but, far from punishing the pair for their lack of discipline, Martinez welcomed it.
"If I'm honest I was glad to see that because the position we are in we need passion," said the Spaniard, whose side will drop back into the bottom three if Aston Villa manage a point at home to Manchester City on Monday night.
"We cannot have players who accept defeat and go through the motions," he added.
"It was a moment of frustration between the players who had the feeling we had thrown away the opportunity to win three points and the reaction from James and Emmerson is something I want to see in the dressing room.
"I think they were demanding a little more from each other and then it became a little argument.
"If everyone could have that passion in the dressing room we would have been a lot more competitive than we were in the first 20 minutes.
"At half-time there was a lot of respect between the two players, who analysed their frustration and used it in a good manner to perform better in the second half.
"I'd rather see that passion on the pitch than players accept defeat and not get hurt when their performance is not what is should be."
The Wigan players' contretemps came midway through the first half, by which time they were already 2-0 down after a second-minute header from Stewart Downing and a smart finish by Luis Suarez.
Liverpool's Uruguay striker went on to claim his hat-trick with a deflected free-kick and a breakaway run early in the second half to become the Premier League's leading scorer with 21.
Wigan have 10 matches left to save their season once again and could probably do without the distraction of a FA Cup quarter-final against Everton next weekend.
Martinez is confident, however, they will bounce back from their poor showing at the DW Stadium, where they have won just twice all season.
"The biggest strength we have as a team is we can go anywhere in the league and are capable of getting points," added the Latics boss.
"But it is clear we need to learn lessons quickly. We can't defend in the manner we did in the first 20 minutes - we need to improve.
"It doesn't matter who you are playing against, if you are that soft it makes it very difficult to win games.
"You can't perform badly and expect to win games and as a team I thought we had too many bad performances defensively.
"Our defensive performance in the first 20 minutes made the game impossible.
"But we are not lower than any other day. You have to congratulate Liverpool and you have to move on.
"We have many things to look forward to, the first being Everton in the FA Cup, and everyone is fighting for places in the team."
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