Mace Windu doubted Qui-Gon Jinn's belief that the Sith had returned to threaten the Jedi and that Anakin Skywalker was the Chosen One of Jedi prophecy. He helped test Anakin's Force abilities, but agreed with Yoda that the boy was too old to be trained.
The answer is no. Mace Windu never once did or said anything that could be construed as him believing he was the Chosen One. The “logic” behind this theory, as I've heard it discussed, is basically: “purple = red + blue = Jedi + Sith = light + dark = balance = chosen one.”
The maverick Jedi Master Qui-Gon Jinn comes before the Jedi High Council, which Windu is a member of, and offers to train Anakin Skywalker, believing that the boy is the Chosen One of Jedi prophecy. Windu and the other Council members decline, deeming Anakin too old and full of fear.
Kenobi exclaimed that Anakin Skywalker was the Chosen One but instead of bringing balance as foretold, he condemned the Force to darkness by joining the Sith.
Obi-Wan Kenobi : Anakin did not take to his new assignment with much enthusiasm. Mace Windu : It's very dangerous, putting them together. I don't think the boy can handle it. I don't trust him.
Giving in to his feelings, Anakin cuts off Mace Windu's hand, giving Palpatine the opportunity to kill him. Anakin regrets this almost instantly, but the thought of Padme was stronger than his remorse This is the moment Anakin officially became Darth Vader and led to serious issues in the Skywalker Saga.
He helped test Anakin's Force abilities, but agreed with Yoda that the boy was too old to be trained. He would always have doubts about Anakin's suitability for the life of a Jedi.
Jedi Master Qui-Gon Jinn met Anakin while he was a slave on the desert planet Tatooine. Qui-Gon got to know Anakin and his mother, Shmi. He discovered that Anakin had no father and had the highest midi-chlorian count ever recorded. This convinced Qui-Gon that Anakin was the Chosen One.
Obi-Wan knew Anakin through and through and understood his overconfidence and arrogance. He knew Anakin would fight with rage and aggression, while he remained cool and calculated. Pairing his understanding of Anakin with his knowledge of Soresu ultimately gave him the upper hand in the battle.
Throughout several quotes and interviews, it's been confirmed that Lucas intended Anakin as the Chosen One in the prequel trilogy, but for some reason never explicitly says as much in the movies.
Plus, whenever talk of the prophecy and Anakin being the Chosen One arose, Mace wouldn't hide his disgust. That made him caustic and belittling towards the young padawan, especially when the Jedi became more integrated into the military as the Clone Wars began.
On top of that Mace Windu told Anakin to wait in the council chambers for other specific reasons such as: He sensed a lot of turmoil in Anakin and he knew Anakin's judgement was clouded.
Like all Jedi, Mace deeply respects Yoda and regards him as the oldest and wisest Jedi of his time. He always obeys the ancient Jedi Master, but as a senior member of the Jedi Council, he is one of the few who dared to question him, though he rarely does so.
Behind the scenes
George Lucas himself has stated that Anakin is the Chosen One and that the prophecy is true, although it had been misinterpreted by the entire Jedi Order. Lucas believes that both the dark side—greed— and the light side—compassion—are integral parts of human beings.
If you are asking about Vader post-Mustafar, then no, she doesn't. Padmé doesn't seem to know Obi-wan cut off his limbs and left him to burn alive, Obi-wan didn't believe Anakin to be alive and Anakin's surgeries were still happening as Padmé died so there was no way for her to know.
David Porter provides some great philosophical context to why Obi-Wan would not regret leaving Anakin alive, as it was obviously “the will of the Force” that Obi-Wan be unable to finish him off, and “the will of the Force” that Anakin live on to kill Sidious himself in the future.
During Palpatine's reign as Emperor, his apprentice's true identity was kept a secret, leaving only a few Star Wars characters aware Anakin was Darth Vader. Besides Obi-Wan, Master Yoda also knew the truth, as well as Ahsoka Tano, R2-D2 and C-3Po, and even Senator Bail Organa.
The book then digs into one of the final scenes in The Phantom Menace, where Anakin watches Qui-Gon's funeral pyre blaze and “feeling a profound sense of loss.” After leaving his mother and the only home he's ever known, Anakin then experienced intense battle and the loss of a person who promised him a better future.
Qui-Gon Jinn was more attuned to the Force than any other prequel Jedi, and that's one reason why Emperor Palpatine needed him dead.
Yoda was reluctant to train Anakin in Episode II because he said Anakin was too old to start training (even though Anakin was still a boy). However, in Episode V, Yoda didn't seem very reluctant to train Luke despite the fact that Luke was a full grown man by that time.
Wait, Did Yoda & Windu Know About Anakin's Secret Relationship With Padme? It's possible that Jedi Masters Yoda and Mace Windu always knew about Anakin Skywalker's secret relationship with Padmé Amidala, and one particular Star Wars: The Clone Wars episode provides proof.
No. Anakin cut Mace's hand because he believed that he needed Palpatine to save Padme and not because Mace didn't follow Jedi's way. Even if it wasn't in the jedi's way, Mace was completely right for what he told to Anakin.