Carragher was particularly critical of the timing of Salah’s comments, coming after Liverpool had conceded a late equaliser to draw 3-3 at Leeds on Saturday evening.
The defending Premier League champions have won just four of their past 15 games in all competitions, going back to September, with Slot increasingly finding himself under pressure.
“He has waited, I think, for a bad result with Liverpool,” Carragher said.
“Everyone connected with the club feels like they are in the gutter and he has chosen this time to go for the manager – and maybe try and get him sacked.
“The one line that stands out for me was ‘thrown under the bus’. He has tried to throw the club under the bus twice in the last 12 months.
“Going after the owners initially, he complained a year ago because they hadn’t given him a contract at the age of 32.
“With the manager, he should be doing all he can to help the club get out of the worst run of results they have had since the 1950s.”
Speaking on BBC Radio 5 Live, former England captain Steph Houghton agreed.
“It probably couldn’t have come at a worse time for Liverpool,” she said.
“That’s what makes us think this is something that was pre-planned. He probably expected to come on at some point [against Leeds], he didn’t, and for Mo Salah that probably hurts his ego.
“But it’s poor from him. I think it does put other people in jeopardy. It’s got us talking about it, which he wants.”
Carragher also called for Salah not to be “obsessed” with his own numbers and help the other players, such as big-money signings Florian Wirtz and Alexander Isak, hoping to emulate his success at the club.
“Some of the criticism has been excessive this season but I will go after Mo Salah when he tries to throw my club under the bus off the pitch and just thinks about himself,” he added.
“The club have made the right decision in him not going abroad with them and whether he will play for the club again I don’t know.
“I hope he does because he is one of the greatest players we have ever had.”