Menga appeared to move his head towards Ryan Christie as they jostled while waiting for a set-piece but although referee Kevin Clancy interjected he took no action.
The point took the Parkhead side top of the Ladbrokes Premiership on goal difference over Hearts.
Afterwards Rodgers, who revealed defender Kristoffer Ajer will need surgery on an eye socket injury sustained in the second half, said: “He should have been sent off. I thought so in real time.
“I thought he gestured his head towards Ryan and when I saw it after the game it was actually more aggressive than I thought and he was very, very lucky to stay on the pitch.
“If he goes down it doesn’t matter if the referee hasn’t seen it. I assume he hasn’t seen it so if he goes down or not it doesn’t matter.
“It wasn’t a great action by the player so you can only assume all the officials haven’t seen it.”
Despite drawing a blank against Livi, Rodgers was pleased to go into the international break top of the table.
The Northern Irishman said: “Livingston test you with how they play with throw-ins, deep free-kicks and I thought we stood up to that really well.
“Then the game opens up a little bit and we had chances and their goalkeeper was man of the match.
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“He made three fantastic saves and I have no complaints. My team has given everything over the last seven weeks and we have to give credit to Livingston.
“A point is a good point here, a positive point for us on the back of all the games we have had.”
Livingston manager Gary Holt was pleased with his side’s performance, saying: “The lads stuck to the plan and we got a deserved point.”
He also claimed Scotland Under-21 international goalkeeper Liam Kelly should be interesting senior boss Alex McLeish.
McLeish will likely have to find a replacement for Celtic keeper Craig Gordon for the Nations League matches against Albania and Israel, after missing the game with a hip and knee injury.
Holt said: “I think I have spoken this week that he should be getting the recognition he deserves.
“You just hope other people are taking note of him because I believe he should be on the fringes or in the squads for people to have a look at him and see what he’s about, expose him to international football.
“I think he’s up there with the youngest and best goalkeepers in the league.”