Mary Lou McDonald says Sinn Féin will provide robust opposition to the new coalition government.
She was nominated as Taoiseach by Donegal TD Pearse Doherty, but Fianna Fáil's Martin exceeded the quota on the first count, winning the support of 93 deputies.
She said: "Today's marriage of convenience is born of necessity, not ambition -- to buy time, to keep others out, to keep others in their place. For the political establishment it's their way or no way.
"I have to tell them that you will no longer get it all your own way, and that day is all over."
Deputy McDonald said Sinn Féin's surge in the February general election proved there was an appetite for change, and a mandate for her party to deliver it.
She observed: "Faced with the prospect of losing their grip on power, Fianna Fáil and Fine Gael have circled the wagons. They excluded Sinn Féin and the voices of more than half a million people from government talks."
She said the Programme for Government agreed between the three parties had "dodged" the key issues of the general election campaign -- and that her party would scrutinise the coalition on them.
The Sinn Féin leader told deputies: "Fairness, equality, decency -- these are the things that will shape Sinn Féin's determination to shape the most effective opposition ever seen in this state."
"No longer is the dream of the the idealistic few. It is now the achievable goal of the many. It's an ambition shared by people from all walks of life, and change pulses through the veins of our nation, fueled by positivity, by hope, and the promise of a better tomorrow."
Galway West's Mairead Farrell seconded Deputy McDonald, warning: "The Greens are repeating the same mistake they made 13 years ago.
"For the political establishment it is their way or no way. The price of having Fianna Fáil and Fine Gael together in Government is too high.
"This wasn't a protest vote. This was a vote driven by kindness, by good will, by compassion, and rooted in common sense."
The alliance between Fianna Fáil, Fine Gael and the Green Party leaves Sinn Féin the largest party by far on the opposition benches.
With 37 seats, Sinn Féin has six times the number of seats of the next largest opposition groups, Labour and the Social Democrats, who have six seats each.
Mary Lou McDonald is the first Sinn Féin president to become leader of the opposition since Éamon de Valera spoke against the Anglo-Irish Treaty in 1922.
A group needs five members or more for speaking rights, which means Solidarity-People Before Profit and three separate groups of independent TDs will also have parliamentary speaking rights.