McGregor was suspended for six months and fined $50,000 (£38,095) for his part in the ugly scenes which marred UFC 229 in October, BBC Sport reported.
Nurmagomedov was banned for nine months and fined $500,000 (£380,950).
"It's just how it played out," said Ireland's McGregor.
In a post on social media he initially said, "I am thankful for the Nevada Athletic Commission's fair assessment and handling of the brawl incident.
"It was not my intention to land the final blow of the night on my opponent's blood relative.
"I look forward to competing again soon."
Russian Nurmagomedov began a brawl by the octagon after beating McGregor, before the Irishman was involved in a bust-up with his opponent's team.
In a second message about the incident, McGregor posted two pictures of him evading a punch before landing one of his own on someone from Nurmagomedov's team.
"Straight left hand inside the attackers jab," he said.
"The final blow of the night at UFC 229 in association with McGregor Sports and Entertainment."
The suspensions for McGregor and Nurmagomedov have been backdated to the date of the Las Vegas fight.
However, Nurmagomedov's suspension could be reduced to six months if he participates in an anti-bullying campaign in Nevada.
If it is, it is not expected he will return to action until after June 5 because the practicing Muslim will honor Ramadan.
Two of Nurmagomedov's team have also been handed suspensions by the Nevada Athletic Commission (NAC).
Both Abubakar Nurmagomedov and Zubaira Tukhugov have been banned for one year and fined $25,000 (£19,030).