Valverde was sacked and replaced by Quique Setien last month after Barcelona were knocked out of the Spanish Super Cup by Atletico Madrid.
Messi and Abidal played together at the Camp Nou for six years between 2007 and 2013 before the latter returned to the club as sporting director in 2018.
"A lot of players were not happy and did not work much and there was also a problem with internal communication," Abidal told Sport on Tuesday.
"The relationship between the coach and dressing room was always good, but there are things that as an ex-player I can sense. I told the club what I thought and we had to make a decision."
Those comments did not go down well with Messi, who responded on social media, asking Abidal to reveal the identities of those responsible for Valverde’s downfall.
"I honestly don't like to do these things, but I think everyone has to be responsible for their tasks and take care of their decisions,” the Barcelona captain wrote on Instagram.
"The players [are responsible for] what happens on the pitch and we are also the first to recognise when we are not playing well.
"Those responsible for the area of sports management must also assume their responsibilities and especially for the decisions they make.
"Finally, I think that when talking about players we should give names, because if not we are all getting dirtied and feeding things that are said but aren't true."
Barcelona have fallen three points behind Real Madrid in the race for the Spanish La Liga title.
