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The actor, 63, was reportedly in possession of the gun when it accidentally fired and killed Halyna.
The 42-year-old director of photography on the movie died shortly after being transported to a hospital in Albuquerque, New Mexico on Thursday afternoon local time, Deadline reports.
In a statement published on his official Twitter page on Friday, Alec wrote: "There are no words to convey my shock and sadness regarding the tragic accident that took the life of Halyna Hutchins, a wife, mother and deeply admired colleague of ours.
"I'm fully cooperating with the police investigation to address how this tragedy occurred and I am in touch with her husband, offering my support to him and his family.
"My heart is broken for her husband, their son, and all who knew and loved Halyna."
The film's director, Joel Souza, 48 was also injured in the freak accident which took place at the Bonanza Creek Ranch in Santa Fe. He has since been discharged after being treated.
Alec was questioned by the police after the accidental shooting and released later on Thursday. Local police have confirmed no charges have been made.
Ever since the devastating news was announced, tributes have been pouring in from Hollywood, with many stars reacting on on social media.
Director James Gunn wrote on Twitter: "My greatest fear is that someone will be fatally hurt on one of my sets. I pray this will never happen. My heart goes out to all of those affected by the tragedy today on Rust, especially Halyna Hutchins & her family."
Actor Joe Manganiello added: "I’m in shock. I was so lucky to have had Halyna Hutchins as my DP on Archenemy. An incredible talent & great person. I can’t believe this could happen in this day and age… gunfire from a prop gun could kill a crew member? What a horrible tragedy. My heart goes out to her family."
Our thoughts are with Halyna's family and friends at this devastating time.
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