"It is correct to say Kristian Nushi did not obtain a visa," the Swiss club told Reuters on Tuesday.
The 31-year-old moved to Switzerland 11 years ago and has played in the country ever since.
Kosovo declared independence from Serbia in February 2008 and has been recognised by at least 100 countries including the United States and 23 of the European Union's 28 members but not by Russia.
Russia, a Serbian ally and United Nations Security Council veto-holder, has so far thwarted Kosovo's hopes of joining the UN.
Kosovo, in turn, has not been able to join European soccer's governing body UEFA which only accepts applications from countries recognised by the UN.
UEFA was not available for immediate comment and St Gallen said they were undecided about whether to lodge a protest.
World soccer's ruling body FIFA agreed in principle in May 2012 to allow its member associations to play friendlies against teams from Kosovo but then put the decision on hold following a protest from the Serbian FA.
In December, it agreed that teams from FIFA member associations could play Kosovo teams at "youth, amateur, women's and club football" level but made no mention of full internationals.
The first leg between St Gallen and Spartak ended in a 1-1 draw.