"As the country's youngest holder of this office he will speak for a new generation of Irishwomen and Irishmen", Mr Kenny told the Dáil.
The government I lead will not be one of left or right.
"As Ireland's national LGBT+ youth service, we are delighted that the new leader of Fine Gael is a gay man, and now a role model for the youth who use our services across the country".
Mr Kenny, from Castlebar, Co Mayo, took over as Fine Gael leader in 2002 after the party suffered near annihilation at the polls.
Earlier this month, Varadkar won leadership of the country's center-right Fine Gael party.
Although long-awaited, the transition of power has been relatively smooth with Leo Varadkar securing resounding support in a leadership contest, including huge backing among his parliamentary colleagues. "It's just part of who I am, it doesn't define me, it is part of my character I suppose".
In a somewhat reluctant and humble valedictory address to the Dail parliament in Dublin, Mr Kenny said he accepted he had made mistakes but that he always had people's best interests at heart.
The popular minister has campaigned for same-sex marriage and liberalising abortion laws. After that, he worked as Fine Gael's minister for transport, tourism, and sport, and then as health minister.
His views are regarded as a centre-right politically due to his approach to socio-economic issues.