The Australian was stood down two weeks ago and last week the director of rugby Chris Chester said the club were "following a process".
Carr, 38, was appointed Tigers head coach on a three-year deal last August after a spell as assistant at NRL side St George Illawarra Dragons.
"The club would like to thank Ryan for his efforts and wish both him and his family all the best for the future," the club said in a statement., external
Carr won seven of his 21 Super League matches in charge and leaves with the Tigers ninth in the 14-team table.
Chester, who has led the team in the past two matches, said last week that they were not "actively looking" for a new head coach.
Owner Martin Jepson said he was "sorry" that Carr had left the club.
"We have had a difficult set of circumstances to deal with and unfortunately not been in a position to provide more information to the fans. However, our actions have been, and always will be, in the best interests of the club," he said.
Jepson confirmed that Jason Netherton would assist Chester and Luke Robinson for the remaining four games of the campaign.
The Tigers visit St Helens on Saturday, looking to build on last week's win at Hull FC.
