Sources tell us officials thoroughly investigated the altercation -- combing through multiple videos and interviewing several witnesses -- and ultimately determined Baer's action's did NOT constitute a crime.

The SF District Attorney's office gave us the following statement ... “After a careful review of the relevant evidence, including multiple videos, statements from several witnesses and the parties themselves, the evidence does not support filing criminal charges.”

TMZ Sports posted video showing Baer in a physical altercation with his wife in a public park -- ultimately knocking her to the ground in a struggle to pry a phone out of her hands. You can hear her screaming, "Oh my god. Help!"

Larry has taken a leave of absence from his job with the Giants while cops and the MLB investigate the situation -- and has expressed remorse.

“I am truly sorry for the pain that I have brought to my wife, children and to the organization. It is not reflective of the kind of a person that I aspire to be, but it happened and I will do whatever it takes to make sure that I never behave in such an inappropriate manner again.”

Baer's wife, Pam, has publicly supported her husband ... calling the incident a non-issue and has insisted they are still happily married.

Baer has been involved with the Giants organization for decades. He was named chief operating officer in May 1996, team president in October 2008 and CEO on January 1, 2012.