Crouch flicked the cap off a Port supporter's head when exiting Adelaide Oval after last Saturday night's fiery clash between the South Australian rivals.
The Power supporter, who was leaning over a boundary fence and yelling abuse at Crows players as they left the field, reported the incident to police.
"It's one I can't comment around, there's an investigation going on at the moment, but Matt has got our full support," Adelaide coach Matthew Nicks told reporters on Wednesday.
SA Police confirmed an investigation was underway following a complaint from the Port supporter.
The Crouch incident follows Port defender Dan Houston being banned for five games for a bump which knocked out Adelaide's Izak Rankine in the match won by the Power.
Port's mascot was struck in the head by a young Crows fan before the game, suffering a bloodied nose. No action was taken because the Adelaide supporter is a minor.
Power forward Willie Rioli was also racially abused online following the game in an act condemned by both clubs.
Crouch was involved in several on-field skirmishes with Port Adelaide players during the game.
In the aftermath, footage shows the Port supporter leaning over a fence and abusing Crows players as they exited through the players' race.
Crouch, not breaking stride, flicked the cap off the fan's head.
"It's a hard one ... the reason that we, I guess, allow fans close to the boundary line is for those kids that just love footy," Nicks said.
"They have idols and they have heroes and they want to come down and get amongst it and (take) selfies and that's the really positive part of our game that we love."
Izak Rankine when knocked out by Dan Houston, who has been banned for five games. (Matt Turner/AAP PHOTOS)
Asked if a proverbial line had been crossed, Nicks replied: "It's just passion for sport.
"It was a big week, a big Showdown," he said.
"We played against a side that were playing for top two, top four, playing for finals; the competition is so tight so it was a huge game.
"And with that comes an intensity, the intensity I think we all enjoy and love.
"When adrenaline is pumping and emotions are high, sometimes we can overstep the line and it can happen to anyone.
"Yeah, some could argue there was a little bit of that on the night."
Post-match, Nicks was captured by television cameras in a brief discussion with Port coach Ken Hinkley.
The Crows coach, referring to the bump which knocked out Rankine, is understood to have told Hinkley that Houston's hit was a 'dog act'.
"Apparently someone was lip-reading ... but I don't need to share everything that we spoke about," Nicks said on Wednesday.
Nicks refused to be drawn on Houston's five-match ban.
"Izak ... we're pleased with how he's recovered, initially it was a bit of a concern on Saturday night," he said.
"But he's in pretty good spirits and that's really the only concern I've had out of it all.
"With respect, I don't care what the punishment is. It doesn't interest me.
"I'm interested in ... our players, their welfare, their health."
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