The Welsh star scored a penalty in Madrid's 2-1 victory, a result which kept the club in the title hunt behind rivals Barcelona.
But Bale earned headlines for snubbing Vazquez, pushing away a congratulatory gesture from the Spanish winger after he successfully slotted home the penalty.
Milla says incidents like that need to be sorted out with Bale and that he has made a "terrible mistake" by snubbing his team-mate's gesture.
"He's had a tantrum, it's a terrible mistake by the player. I imagine that Bale has already asked Lucas Vazquez for forgiveness. Any football team, professional or not, cannot afford to project a bad image," Milla told Goal.
"A team works with shared values and making that gesture in front of so many people goes against the solidarity of everything. The problem is that it's one mistake after another – we already saw it at the Wanda Metropolitano.
"It gives the impression that he's a player who doesn't connect with his team-mates and that's where Real Madrid have a problem. They want to project an image of a superior team but then a player does something like that.
"I think it's unfortunate and I understand that young players make mistakes, but it's very bad for the image of the club because it shows a lack of respect for your team-mates. Especially Vazquez.
"It's a subject that needs to be fixed from within. There needs to be more awareness."
Marcelo, who has a locker next to Bale, has previously claimed the 29-year-old only speaks English, suggesting it's difficult to communicate with him, while Thibaut Courtois has insisted that Bale skipped a team meal as it was "too late at night".
Bale has been with Madrid since 2013 and has scored 101 goals in 220 appearances for the club. He's also collected four Champions League titles, a Liga crown and a Copa del Rey during his time in the Spanish capital.
Next up for Real Madrid are a pair of Clasicos as they take on Barca in the Copa del Rey on Wednesday before facing off in league play on Saturday.