There is no explicit indication in Big Hero 6 of the origin of the fire.
In Big Hero 6, Callaghan never discusses the fire except right after he is revealed to not only have survived the fire, but ...
to be the villain of the movie.
At this point, the only dialog is:
Hiro: Pro ... Professor Callaghan. The fire ... You died.
Callaghan: No. I had your microbots.
Brief interlude showing Callaghan putting on the transmitter and forming a microbot shell around himself.
Hiro: But, Tadashi. You just let him die.
Callaghan: Give me the mask, Hiro.
Hiro: He went in there to save you!
Callaghan: That was his mistake!
After this, Hiro orders Baymax to destroy Callaghan.
However, earlier in the movie, Hiro theorizes that the "guy in the mask" stole his microbots and set the fire to cover his tracks. While Callaghan does not confirm this, this is the only explanation we have for the fire in Big Hero 6.
Additionally, Disney's cast announcement does imply that Callaghan (called Yokai as the villain) started the fire. If that is the case, than Hiro's theory is most likely correct.
YOKAI is the silent masked man behind the terrible tragedy that strikes San Fransokyo, turning Hiro’s world upside down. As a result, Hiro transforms a group of nerds into a team of high-tech crime fighters—Big Hero 6—with one mission: Track down Yokai and bring him to justice.
Finally, Big Hero 6: The Series picks up after the events of the movie. As of October 2018, Callaghan has only appeared in the episode "Mini-Max," where he is remorseful for what he did in the movie. However, the only explanation he gives is:
Callaghan: I never set out to hurt anyone. What happened to Tadashi, because of me, I know it's not enough, but I'm sorry.
Callaghan is never stated to have started the fire in the Big Hero 6 movie or TV show, so we cannot prove that he did. However, the cast announcement and Hiro's hypothesis imply that he may have done so, or at least did not rescue Tadashi when he had the chance. Because the connection could have been further explained and hasn't, I would assume that Disney's intention is to keep it ambiguous.