The actor, who was 49, was diagnosed with ALS - also known as motor neurone disease - five and a half years ago.
His family said in a statement on Instagram that Mitchell was "an inspirational work of art to all the hearts he touched".
Mitchell appeared in a variety of TV shows and films, including Captain Marvel, in which he played the titular character's father, Joseph Danvers.
Born in Toronto, Canada, Mitchell was married to actress Susan May Pratt and the pair had two children together.
He revealed that he was suffering from ALS in an interview with People magazine in 2020, and said he had been using a powered wheelchair.
In the post on Instagram, his family paid tribute to him as "a hope seeker, daydreamer, dream believer, soccer player, beach walker, bear hugger... and more than anything else, a proud father".
