Team boss Christian Horner was quite pointed in his criticism of Verstappen - who has had a series of mishaps this season - on British television after the session.
A virtual safety auto period ensued, and Ricciardo was the main beneficiary as he emerged five seconds clear of Vettel with just three laps to go.
Hamilton, who himself remains in negotiations with Mercedes over his new deal, added: "He should definitely have a contract on the table but I was talking to him and I don't know if he has his options planned out".
Ricciardo is the key cog in this year's driver market and has raised his value with two highly-praised wins in China and Monte Carlo.
Fernando Alonso termed the 2018 Monaco Grand Prix as the most boring race ever.
"I felt it (offering the shoe) would have been controversial, so best not". Lewis Hamilton put the lead Mercedes on third on the grid.
Hartley's Toro Rosso team opted to pit him earlier than most - on lap 14 - but he was handed a five-second time penalty for speeding in the lane.
"I think it was definitely my best weekend and the most satisfying, " Ricciardo replied when asked if his win in Monaco was his most satisfying.More news: President Trump, Kim Jong Un summit needs a plan
"Daniel did a great job today, so super happy for him, but ultimately we were all turned down and just cruising around, making sure we get to the end. We knew that would be the case". And then maybe we're a nice little outside chance. Daniel had the answers at all times. He in a vehicle that is capable of winning this grand prix and that will hurt him even more. Valtteri Bottas (Mercedes) finished in fifth place, while Esteban Ocon (Force India) finished in sixth, ahead of Pierre Gasly (Torro Rosso).
"The tyre's bad", said Hamilton, who led Bottas in to pit for new hyper-softs.
Ricciardo was told by his race engineer the problem would not improve, with the Australian left to try to cope with his vehicle with still over half of the race left to go.
"If at some stage Red Bull get an engine they're going to be even closer in the mix with us, and there is still a long, long way to go [this year] obviously".
"Monaco has the biggest build-up and is the most special race of the season and it is just a shame that the race isn't as exciting as the whole spectacle and the track actually is".
"At Ferrari I would imagine Kimi [Raikkonen] is going to want to continue and why not?"
Vettel said his Ferrari was quick enough to stay with Ricciardo at times before the problem, but believes his tire wear was higher than the Red Bull, which resulted in his challenge fading away after a late Virtual Safety Car period.