Pressure has been mounting on Jackson to either resign his seat or to not seek re-election.
The visit was "really emotional," Kennedy said.
Kennedy added he could tell Jackson had been "dealing with deep depression."
Kennedy stressed that Jackson is an "inspiration" for those suffering from mental health ailments and that he's "showing people this is a serious issue that should be dealt with like any other medical condition."
Kennedy said he advised Jackson to be transparent about the illness and the treatment with his constituents "as if he had cancer and [show] this is just as deadly."
As a Congressman, Kennedy was a major sponsor of the 2008 Mental Health Parity Act, requiring most health care plans to provide coverage for mental illnesses comparable to what is offered for physical illnesses.
Kennedy served in Congress for 16 years and is the son of late Sen. Edward Kennedy (D-Mass.) and nephew of President John F. Kennedy.
Jackson has served in Congress since 1995 and is the son of civil rights advocate and former presidential candidate the Rev. Jesse Jackson.
Earlier this week, a Mayo Clinic blog post revealed that Jackson was being treated at the Minnesota facility for bipolar disorder possibly tied to gastric bypass surgery, which he underwent in 2004.
Jackson started medical leave in June, although his office did not specify why and waited about two weeks before notifying his constituents and the media.
After rampant speculation, his office finally divulged that he was being treated for a mood disorder, although the specific type had not been made public until the posting from the Mayo Clinic.
Vice President Joe Biden waits to conduct a mock swearing-in ceremony with Sen. Brian Schatz, D-Hawaii, in the Capitol's Old Senate Chamber, December 2, 2014. Schatz was sworn in to serve the remainder of his term since he was appointed to the seat after Sen. Daniel Inouye, D-Hawaii, passed away.