The sixth seed beat Czech Muchova 6-4, 7-5, at Arthur Ashe Stadium to reach the championship match at Flushing Meadows for the first time and continue her terrific season.
After winning the first set and taking the opening game of the second, the match was halted when four protesters began shouting from the stands.
Three were escorted out of the venue without further incident but a fourth "affixed their bare feet to the floor" of the stadium, the USTA said.
"It was done in a peaceful way, so I can't get too mad at it," Gauff said. "Obviously I don't want it to happen when I'm winning up 6-4, 1-0, and I wanted the momentum to keep going.
"But hey, if that's what they felt they needed to do to get their voices heard, I can't really get upset at it."
The teenager is no stranger to social activism, speaking out at a protest following the death of George Floyd, an unarmed Black man killed in police custody in the United States, three years ago.
While she said she was not clear on the precise nature of Thursday's environmental protest, Gauff said she believed that climate change was real and supports "preaching about what you feel and what you believe in".
"Would I prefer it not happening in my match? 100%, yeah," said Gauff. "I'm not going to sit here and lie. But it is what it is."
It was not the first time this season that activists have interrupted a tennis major, after 'Just Stop Oil' protesters disrupted play at Wimbledon.
