Rose has been one of three hosts of CBS This Morning since 2012 and is also a contributor to 60 Minutes.
The network giant suspended airings of Rose's nightly interview shows after the Washington Post published an article on Monday about women who said he groped and walked around naked in front of them.
CBS had already suspended him.
Rose's co-hosts on the morning show, Gayle King and Norah O'Donnell, were sharply critical of their colleague on Tuesday.
King said the allegations did not fit the Rose she knew, but that 'I'm clearly on the side of the women who have been very hurt and damaged by this.'
In response, Rose has said that he doesn’t 'believe that all of these allegations are accurate' but that he would like to 'deeply apologise for my inappropriate behaviour'. He added, 'I am greatly embarrassed. I have behaved insensitively at times, and I accept responsibility for that'.