Melvin Gordon issued an apology to his Broncos teammates and the entire city of Denver for his DUI arrest earlier this month ... saying straight-up, "I am sorry."
The running back was accused of drinking and driving back on Oct. 13 ... after cops say they pulled him over for doing 71 MPH in a 35.
In the police report, obtained by TMZ Sports, the arresting officer claimed Gordon had slurred speech, watery eyes and reeked of booze.
Gordon was eventually hit with 2 DUI charges and a speeding violation over it all .