Mr Martin met with members of the Finucane family earlier this week, and restated his and his Government’s support for the holding of such an inquiry, given the collusion of British forces in Mr Finucane’s death 31 years ago.
The timing of the call was delayed because of the Cabinet meeting in Dublin and the announcement of the country’s plan for reopening.
Mr Martin had engaged with Mr Johnson on the matter earlier this week, ahead of the court deadline of Monday, by which time the British government must respond to a UK Supreme Court ruling made last year.
Mr Martin was “unequivocal and unambiguous” in giving the family not only his personal support but also the support of the Irish Government, Mr Finucane’s son and Sinn Féin MP John Finucane said.
“He didn’t need us to explain to him the significance and importance and the need for an inquiry into the murder of my father, and he committed to, and I understand has already engaged with Boris Johnson on this issue,” said Mr Finucane.
Mr Martin said that the British government “should and must” hold a public inquiry into the killing.
Pat Finucane, 39, was shot by loyalist paramilitaries in front of his family in 1989 amid claims of collusion with state security forces.
In February last year, the British Supreme Court ruled that investigations into the fatal shooting of the solicitor have not been effective, and fell short of international human rights standards.
Northern Ireland secretary Brandon Lewis has indicated that a decision over how the UK government will proceed will be announced.
Mr Finucane said a fully independent public judicial inquiry was the “only mechanism” that could “fully grasp all of the issues” that led to his father’s murder.
Last week it emerged that four of Northern Ireland’s political parties had united in calling for a public inquiry into the murder of Mr Finucane.