On Tuesday, Schulz said he hoped his decision to resign would result in party members focusing on the coalition treaty rather than personnel issues.
Speaking to Germany's Funke media group, Gabriel said he felt that while "in the eyes of the public I have clearly carried out my duties very well and successfully ..."
The coalition deal signed with Merkel's conservatives could be in jeopardy as the battle for the SPD leadership continues to play out.
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The SPD's 464,000 members vote in a postal ballot beginning on February 20 on whether the party should go ahead with the agreement its leaders clinched last week to renew their power sharing alliance with Merkel's conservative bloc. Schulz proposed the head of SPD's parliamentary fraction, Andrea Nahles, as his successor, but it's now up to the party's congress to decide on its leadership. If members reject the coalition pact, a new German election looks the most likely option.
Deeply divided over the coalition deal and the division of ministerial posts, and facing a slump in opinion polls, SPD leaders are trying to convince 464,000 party members to back the deal in a ballot on which Merkel's fourth term hangs. "End the lack of discipline in the SPD!" one of the party's deputy leaders, Ralf Stegner, told the Rhein-Neckar-Zeitung.
Ms Nahles was named by Mr Schulz as his chosen successor last week. The party's youth wing is already campaigning for a no-vote as it fears that another grand coalition with Mrs Merkel's conservatives will further erode the party's identity. The former European Parliament president said last Wednesday that he planned to become Germany's next foreign minister - a risky move given that, after the election, he had explicitly ruled out entering Merkel's next cabinet.
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Her decision to wear American fashion brands, such as Everlane, is also a way Markle is stepping outside the traditional lines. Lord Lieutenant Frank Ross joined them on their visit to Edinburgh Castle just in time for the firing of the One O'Clock Gun.
In a cartoon published on Tuesday, the Sueddeutsche Zeitung daily showed Nahles with a whip riding an SPD snail.
Scholz played up the SPD's achievements in coalition negotiations with Merkel's Christian Democrats (CDU) and her Bavarian allies, at which the centre-left party won the powerful finance ministry to the frustration of many in the CDU.
The chairman of the Association of Social Democratic Lawyers in the SPD, Harald Baumann-Hasske, also brought up the legality of appointing a new chair, per the party's charter.
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The development followed Schulz's announcement Friday that he was abandoning plans to become foreign minister in the new government.