Mr Ryan received 12 votes in support and 115 votes in opposition.
Mr Varadkar will travel to the Aras an Uachtarain to resign from office, however he will continue to carry out his duties as Taoiseach in a caretaker capacity until a new taoiseach is appointed.
He told the Dail: "The responsibility is on all of us to ensure we provide good government, and indeed good opposition.
"I think the onus is on those who have made enormous promises of change to the people during this election who are entrusted with that mandate to bring a programme of government for approval.
"If they cannot, they should say so and be upfront and honest about their failures and the empty promises they made."
He said he will travel to Washington DC for the traditional St Patrick's Day events.
Mr Martin accused Sinn Fein leader Mary Lou McDonald of using "populist tropes" as she accused them of failing to speak to her in government formation talks.
"Every party and deputy elected to this house has a right and a duty to represent the mandate they received. In order to deliver functioning government, compromises have to be made and this is something we have been attacked for.
"Demanding that others stay quiet as others abandon their core beliefs in order to grab power is notsomething we agree with.
"Whether or not you agree with Fianna Fail, we fought this election based on a clear set of principles and policies and that includes legitimate behaviour in a democratic republic."
Mr Martin says he rejects Sinn Fein's manifesto promises of increasing taxes on businesses and too many incompatibilities exist between the two parties.
Ms McDonald told the Dail: "Today for the first time a nominee other than Fianna Fail or Fine Gael has garnered the greatest number of votes. Perhaps that makes us winners again.
"I still we still live rent free in Micheal Martin's narrow and bitter mind.
"Change means a secure roof over people's heads, not having their adult children living in the box rooms.