Athens "should show some fairness to the people in Germany and the eurozone who have demonstrated solidarity" towards Greece, Gabriel told reporters after Greece's radical new Prime Minister Alexis Tsipras warned that it would no longer blindly submit to the EU and demanded a "fair" deal with its creditors.
"Our aim must be to keep Greece in the eurozone," Gabriel said.
"But there must be fairness towards our own population," he continued.
"Greece cannot simply pick and choose what it does and does not want to do and expect its neighbours to step into the breach," Gabriel insisted.
It was not right that a country should not push through reforms and expect those who do not work and live in that country to pick up the tab, the minister said.
Meanwhile, Bundesbank Executive Board Member Joachim Nagel told economic newspaper Handelsblatt that "if the continuation of the programme of aid for Greece is called into question... Greek banks would lose access to central bank funds."
"It would have fatal consequences for the Greek financial system," Nagel, said.
Greece's partners in the eurozone have loaned it a total of nearly 200 billion euros ($227 billion) at various stages, via the European Financial Stability Facility (EFSF) and bilateral loans.
Germany, as Europe's biggest economy and effective paymaster, has lent it the most.
In his first speech to the new Greek cabinet, Tsipras said he wanted a "viable, mutually beneficial solution" to "humanitarian disaster" he claimed his country has suffered as a result of the austerity imposed by it creditors.
The 40-year-old premier insisted Greece's new leaders were no longer willing to bow to the "politics of submission", in a clear swipe at Brussels and the International Monetary Fund.
"Our people are suffering and demand respect.... We must bleed to defend their dignity," Tsipras said.
The new government's radical anti-austerity agenda has alarmed financial markets, reviving fears that Greece could crash out of the eurozone.
The EU has set the end of February as the deadline for Greece to carry out more reforms in return for a €7-billion tranche of financial aid from the 28-member bloc and the International Monetary Fund.
Tsipras has dismissed the deadline as meaningless.