McDonald was suspended eight games without pay for violating the NFL Policy and Program for Substances of Abuse earlier this year.
McDonald pleaded no contest in January to one charge of "wet reckless" (essentially a reduced form of DUI). He was sentenced to 36 months probation, and to serve 200 hours of community service.
Miami signed the free agent to a one-year deal this spring. The team then signed McDonald to a four-year extension worth $24 million including $10 million guaranteed in September.
McDonald started all 53 games he played for the Rams during his four-year career, compiling 4 interceptions, 18 pass deflections, 2 fumble recoveries, 1 forced fumble, 5 sacks and 218 solo tackles. He was selected by the Rams in the third round of the 2013 NFL draft.
Also Saturday, the Dolphins placed starting right tackle Ja'Wuan James on injured reserve with a hamstring injury, ending his season.
James was hurt in last Sunday night's loss to Oakland. Jesse Davis, who has two career starts, will take over at right tackle Monday night at Carolina, ending James' streak of 25 consecutive starts.
Information from The Associated Press was used in this report.