Tensions are also reported between him and Sigmar Gabriel, an SPD colleague who is now foreign minister.
The new agreement between the two parties has seen the CDU give the SPD three prestigious ministerial posts of finance, foreign affairs and labour.
A deal to govern with Chancellor Angela Merkel's conservatives still has to be put to a vote of SPD members.
Andrea Nahles, a plain-speaking 47-year-old former labor minister with a left-wing slant and strong oratory skills, will become Schulz's successor.
Merkel said on Sunday that if SPD members rejected the coalition agreement, Germany would probably hold a new election.
Leader of the SPD group in Thurianga, Matthias Hey, said: "Our mood regarding what is happening in Berlin now is down in the cellar". He told German news agency, "Die Welt", that "there no statutory basis exists in the SPD's rules that allows for the appointment of a new party leader". SPD members will vote on the coalition deal next month - the result is expected on 4 March.
Mr Schulz also faced anger from across the SPD after taking the foreign ministry job, having previously insisted he would never to serve in a government led by Mrs Merkel. "No more indiscipline in the SPD", said Ralf Stegner, the regional SPD leader in Schleswig-Holstein.
In a cartoon published on Tuesday, the Sueddeutsche Zeitung daily showed Nahles with a whip riding an SPD snail.
Nahles, a former leader of the SPD's youth wing, should work for "a new social idea" rather than invest in an alliance with dwindling support, added Loetzsch, who is serving her fifth term in the Bundestag. Media have speculated that one option might be Katarina Barley, a former SPD general secretary and family minister, or SPD veteran Thomas Oppermann. Support for Merkel's bloc slumped to 29.5 percent in an INSA poll of 2,608 people taken February 9-12, from the 32.9 percent of the votes it got in the September 24 federal election.
The turmoil in the SPD can only distract from criticism of Merkel from within her own party after she handed the foreign and finance ministries to the SPD to secure the coalition deal. Merkel's conservative bloc was also down 1 point at 29.5 percent.
Martin Schulz formally resigned as leader of Germany's Social Democrats, drawing the curtain on his career in the upper echelon of domestic politics less than a year after he took over the party.
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 past year.