The Dutch driver, who still finished third in a race won by Mexican team-mate Sergio Perez, refused to let the matter go despite later taking back the place after a safety car deployment.
"I don't understand how he can keep the position when he's damaging my car," he said over the radio.
When the Dutchman regained the place Verstappen was congratulated pointedly by the team for a "lovely job, without contact as well".
Verstappen and Russell exchanged words after the race, television microphones picking up Verstappen hinting at future revenge and calling Russell a "dickhead" as the Briton walked away.
"I respect all the drivers a lot and it's a bit of common sense as well, what you do on the first lap," Verstappen told Sky Sports.
"They (Mercedes) are off the pace and to risk that much in lap one is not very rewarding because I will get him anyway within a few laps.
"He tells me 'yeah, I got cold tyres'. We all have cold tyres, that's not an excuse. I don't like it when somebody just has that kind of reaction - 'I didn't do anything wrong' - and just walks off.
"I would try to fight but not run into someone, that's not what you want to do. That's something maybe you also learn over time."
Russell, who finished fourth for a team playing catch-up after a disappointing start to a season so far dominated by Red Bull, defended himself separately and said he was there to fight.
"There was a lot of Fs and whatnot from his side," he said. "I was surprised he was so angry to be honest. He still finished the race in P3.
"He's got enough experience to know that if you're trying to overtake a guy on the outside there's a risk the guy on the inside is going to run wide into you.
"I don't think anything would have been any different had the positions been reversed."
Red Bull boss Christian Horner said the hole in the bodywork had cost Verstappen three-quarters of a second in lap time at least, leaving aside other complications with cooling going out of control.
"I think he voiced his opinion to George, it was probably a good idea he (Russell) left his crash helmet on," he quipped of the post-race exchange.
Verstappen starts from the front row in Sunday's main grand prix, alongside Ferrari's pole-sitter Charles Leclerc, with Russell lining up 11th.
Verstappen slams sprint format: "Just scrap the whole thing"
In Baku, the weekend structure was changed with there being a qualifying session on Friday for the main race, a qualifying session for the sprint race on Saturday and the sprint race that followed later that day not having any impact on Sunday's action.
The new standalone sprint format had its fans on Saturday but Red Bull's double world champion was not among them.
"Just scrap the whole thing," he said after finishing third in the 100km race, the first of six scheduled for 2023's record 23-round season.
"I think it's just important to go back to what we have, and make sure that every team can fight for a win, that's what we have to try and aim for and (not) try to implement all this kind of artificial excitement.
"It's not proper racing, it’s more like gambling. I think I will have more success in Vegas if I go to the casino. So yeah, I like racing, I'm a pure racer and I think this is more for the show."
