Mike Nesbitt made the proposal during his speech at his party's annual conference on Saturday.
The Maze peace centre proposal, Mr Nesbitt said, was "wrong".
He said the plan had "put too much emphasis on the victim-makers and trampled on the sensitivities of those they hurt".
He added: "Our focus must always be on those who were given no choice about becoming a victim."
The former broadcaster said his party "forced" First Minister Peter Robinson into a u-turn on the Maze project.
"And we did it without a riot, without street protests, without so much as a white line protest. Brains, not brawn," said Mr Nesbitt.
He continued: "What is missing from the Maze debate is an alternative to a peace centre at that most controversial venue.
"I offer that alternative, an alternative that addresses the hidden legacy of the Troubles - poor mental health and wellbeing."
He added: "Having spent 45 years creating more post-traumatic stress disorder sufferers per head than just about any other country in the world, let us build a legacy project that cures the problem.
Let us commit to helping restore good mental health and wellbeing to our people, not least the young who self-harm to the point of suicide.
"And be in no doubt, people born after the ceasefires are among those suffering the trauma of the legacy of our Troubles," he added.
"Let us do it for them."
Mr Nesbitt suggested the new facility could be based at Ormiston House in east Belfast.
The leader also made a bid for his party, which has one minister in the Executive, to assume the education department after the next Stormont elections.
He told the delegates: "Education is our number one priority.
"Too many of our young people leave school without the qualifications they need to build a future.
"In too many cases, that is not their fault - it is ours."
He added: "If you believe, as I do, that every child is unique - that inside every child is a spark of ability, creativity and talent that may find its voice equally in the science laboratories, or the music room, or on the sports fields, or at the computer, then we must cherish all those talents and provide a curriculum that develops every scrap of talent in every child, and provides space for all to flourish.
The legacy of the decision to abolish the 11-Plus without an agreed way forward continues to poison education in Northern Ireland, as Sinn Féin remain fixated with the signs hanging over the entrance to schools.
"An Ulster Unionist Education Minister would refocus on the pupils and parents and teachers walking in and out of the door."
Mr Nesbitt said an Ulster Unionist education minister would not bring back the 11-Plus.
He added: "I want a single education system.
"I know others have said the same, but no one has done anything about it.
"Give us the ministry and we will start the process on day one."
The leader also called for a new covenant with the people of Northern Ireland.
He said: "A covenant that recognises that we can do better for all our people by shaping a fairer education system, a stronger economy, better housing and a health service not only free at the point of delivery but with delivery points that are accessible and appropriate to the needs of our people."