Hamilton's mood erupted in qualifying, when he suspected Rosberg made a deliberate mistake under braking for the Mirabeau to spoil his run at pole.
Afterwards, the 29-year-old said he is "not friends" with the German, and after Rosberg won the race, Hamilton refused even to make eye-contact.
"That Lewis did not shake his hand is not in order," boss Niki Lauda said.
Another world champion, John Surtees, agreed: "I didn't like what I saw and heard from Monaco."
Hamilton even came under attack from his home town of Stevenage, having derisively compared the UK borough with Rosberg's extravagant Monaco upbringing.
Councillors said Hamilton calling Stevenage "not great" was "unfair".
Hamilton has now moved to ease his anger with Rosberg, after declaring in Monaco that despite their long history, they are "not friends".
On Facebook and Twitter late on Friday, he posted a photo of the teenaged pair riding unicycles, with Hamilton's arm around Rosberg.
"We've been friends a long time and as friends we have our ups and downs," Hamilton wrote. "Today we spoke and we're cool, still friends, no problem."