Turkey sparked a months-long row after it twice blocked German lawmakers from visiting the 250 Bundeswehr troops that were stationed at the southern Turkish base at Incirlik.
The German foreign ministry insisted that all sides, including North Atlantic Treaty Organisation, remained in talks on setting a new date for a visit.
"Under these conditions, I see no way to extend the mandate", said Hellmich, adding that Turkey's decision had been conveyed to the German Foreign Ministry a day earlier on Thursday.
Germany's armed forces are under parliamentary control and Berlin says the lawmakers must have access to its soldiers.
Turkey asked the legislators to postpone a scheduled visit next Monday to a North Atlantic Treaty Organisation base in Konya, the German foreign ministry said, adding that it regretted the decision. Germany relocated its troops from Incirlik after repeated attempts to gain lawmaker access. "The government, especially Chancellor Angela Merkel, must now take the necessary steps to ensure lawmakers can soon visit the soldiers in Konya", said Rainer Arnold, defence spokesperson for the Social Democrats, the junior coalition party in the government.More news: Linfield to probe bottle incident involving Celtic's Griffiths
Relations between Turkey and Germany, home to millions of ethnic Turks, have been badly strained, especially since the failed coup attempt nearly one year ago against Turkish President Recep Tayyip Erdogan.
Merkel's criticisms of Ankara have been muted, partly because she needs its cooperation in upholding a deal whereby Turkey prevents migrants and refugees, many fleeing Syria's civil war, from leaving its shores for the European Union.
"If parliament can't visit, the German army cannot remain in Konya", said Thomas Oppermann, a senior SPD lawmaker in a statement.
"This whole issue is unfortunate, very unfortunate", Merkel told public broadcaster ARD, adding that more talks were needed to find a solution, also with the help of North Atlantic Treaty Organisation.