Rangnick said he was keen to include the striker for the trip to Villa, but Martial turned down the opportunity.
"He didn't want to be in the squad," Rangnick said.
"He would have been in the squad normally, but he didn't want to. That is the reason why he was not travelling with us yesterday."
However, Martial refuted his manager's claims in an Instagram post later on Saturday, saying he would never reject a chance to play for United while he is still at the club.
"I will never refuse to play for Man United. I've been here for 7 years and I never disrespected and will never disrespect the club and fans," he said.
Martial has not featured since the 3-2 win over Arsenal on Dec. 2. and has told the club he wants to leave in January.
Sources have told ESPN that the club will only consider offers that cover Martial's wages and include a loan fee. The 26-year-old is under contract at Old Trafford until 2024 with the option of another year.
Meanwhile, Rangnick said forward Cristiano Ronaldo was not part of the squad at Villa because of injury. The 36-year-old missed the FA Cup tie with Villa on Monday because of a hip problem and has not fully recovered despite returning to training last week.
"He couldn't train yesterday [Friday] and speaking after the training session it makes no sense to travel without knowing if he could play," Rangnick said.