Mr Schulz's decision to stand down was made after he lost the confidence of the party membership but a row immediately broke out over attempts to install Andrea Nahles as new leader without a contest.
Schulz said last week he would quit to allow the party to regroup and he recommended Nahles as leader.
In his second resignation in several days, the now-former head of the SPD Schulz gave his support to parliamentary group leader Andrea Nahles in an upcoming leadership election that will take place on April 22nd.
The SPD has slid into disarray since its leaders struck the coalition deal, blighted by bitter divisions over whether to join the coalition, a loss of confidence in outgoing leader Martin Schulz, and discontent over the succession process.
Results are due on March 4 and "no" vote would nearly certainly mean new elections and yet more uncertainty in Germany.
"I depart this office without bitterness or resentment", he said, nearly a year after he was elected as SPD party head with 100 percent of the vote.
Scholz is widely expected to become finance minister and vice chancellor in the new government if the coalition agreement is approved.
Many within the SPD harbour misgivings about sharing power with Ms Merkel, believing the party should rebuild in opposition after suffering its worst result in last September's election since Germany became a federal republic in 1949.More news: Prince Harry and Meghan Markle delight fans in Scotland
Analysts said Mr Kühnert, who said he has been inspired by the campaign of Bernie Sanders in the United States, did not yet have enough sway over the rump membership of the party to win the vote on the coalition.
That leaves open who from within the SPD may take up that post.
Handing over the leadership of the SPD may not be as simple as many in the party would like as resistance to Nahles is growing.
Malu Dreyer, regional prime minister of Rhineland-Palatinate, said: "The SPD can not remain leaderless". Mr Schulz on Friday abandoned his plan to become Germany's next foreign minister after the SPD's youth wing accused him of securing a favourable position for himself.
Ralf Stegner, the regional SPD leader in Schleswig-Holstein, said: "Each of us would be well advised to put the interest of the party and the country before their own ambitions". Ms Merkel's CDU secured five ministries, with the CSU winning a further three.
German Foreign Minister Gabriel lashed out at Martin Schulz, saying his party colleague had broken a promise not to serve in Chancellor Angela Merkel's new cabinet.
The SPD originally wanted to reinvent itself in opposition but reconsidered when Merkel's attempt to form an alliance with two smaller parties failed late a year ago.