Martial received treatment on the pitch after a clash of heads with Hornets full-back Daryl Janmaat, and was then on the receiving end of a heavy challenge from Miguel Britos in the build-up to the home side's opening goal.
The France international was unable to continue after the incident and was replaced by Ashley Young in the 38th minute.
Speaking at the post-match press conference, Mourinho said he was unaware of the severity of Martial's injury but insisted that United should have been awarded a free-kick for Britos's challenge.
"I had to be on the game at that time and not worrying with Anthony," the Portuguese coach told United's official website.
"I have not had time to see about the situation. It wasn't an action to go to hospital. It was an action for a free-kick against Watford and obviously not a goal."
When pressed about the injury later on in the press conference, Mourinho responded: "I don't know, I promise I don't know. You are asking about things I cannot tell you.
"I didn't know why he came off, whether he was injured in his knee, his ankle or his head. I don't know.
"I just know, in that moment, he is not in a condition to carry on. I don't know why. I just know I had to change. I don't know yet what happened."
Martial has featured in all five of United's league fixtures this season and also started the 1-0 Europa League defeat against Feyenoord last week.
He could be a doubt for United's League Cup third-round trip to Northampton Town on 21 September.
The Red Devils' next Premier League assignment sees them taking on champions Leicester City at Old Trafford on 24 September.