Schulz's SPD are 14 percentage points behind Merkel's Christian Democrats (CDU) and their Bavaria-based Christian Social Union sister party three months before a national election on September 24.
The party leader laid down gay marriage as a condition for any future government coalition involving the SPD.
Ireland's 2015 marriage equality vote was a rude awakening for many Germans, prompting much soul-searching about how the country had been overtaken on a social issue "even by Catholic Ireland". Volker Kauder, leader of the parliamentary group of the ruling CDU faction, called Tuesday for CDU members voting for and against the law to show respect for each other's position, according to NTV.
Earlier on Sunday, ex-Chancellor Gerhard Schroeder, the last Social Democrat to lead Germany, told delegates the party could still win, reminding them of the SPD's fightback to run Merkel close in the 2005 election - the last one he fought.
"Equal rights for all people is part of our constitution", he added.
"We are living in a time of upheaval".
A demand for an urgent vote before the Bundestag starts its summer break Friday evening was echoed by many other MPs.
"Please don't forget who has withheld #Ehefueralle from you for 12 years!" CDU parliamentarian Stefan Kaufmann wrote on twitter. The far-left Left party were steady at 9. The left-leaning Die Linke party has long supported full equality.
Merkel's party, the Christian Democratic Union, has long opposed gay marriage.
A person takes part in the 39th Gay Pride Parade in Mexico City, capital of Mexico, on June 24, 2017.
Trouble for Merkel's sister party?
Peter Ramsauer, a former cabinet minister and a legislator for the Christian Social Union in Bavaria, rejected outright the need for a vote. Why is it suddenly possible that same-sex couples could be able to marry by the end of the week? "But the (party) leadership should be wary of destroying the last conservative values". "This is not the case in homosexual relationships", the CSU's whip in Germany's parliament, Gerda Hasselfeldt, told ARD television.
The SPD will be hoping its campaign programme will help it regain ground but Schulz himself has warned of a "rocky road" ahead after his party suffered heavy losses in three straight state elections viewed as dress rehearsals ahead of the September vote.
The CDU has additionally formed a coalition with the Greens and FDP in Schleswig-Holstein, indicating its willingness to explore different coalition options. "Families with children will lose special protections". In particular, he noted that the alliance blocked the legislation initiatives regarding same-sex marriage and granting Lesbian Gay Bisexual and Transgender (LGBT) couples the right to adopt children. He has criticised Germany's large trade surplus with the United States.