Swiss coach Murat Yakin and Swiss Football Federation (SFV) communications chief Adrian Arnold declared Xhaka, who was forced out of the World Cup qualifier against Greece, exercised his right not to receive the vaccine.
Xhaka has faced stiff criticism in the Swiss media for snubbing the opportunity to bolster his body's defences as the pandemic continues to surge.
"We can simply give recommendations to the players," Yakin said. "It is his decision not to be vaccinated and we must respect it. He's the captain, but he's also a man. It is his own rights."
According to Arnold, Xhaka was the only member of the Switzerland squad not to have had the jab or have already recovered from the virus.
The Arsenal player's case comes in the week that the SFV has written to its 1,400 clubs and 300,000 members urging them to be vaccinated. It also follows Xhaka's red card for a dangerous tackle on Gunners duty in a 5-0 defeat against Manchester City.
Arnold said, quoted by several Swiss media: "Granit Xhaka was not vaccinated. He's a player who isn't vaccinated. We left this up to each player, it's a personal decision of each player – just like any other person in Switzerland.
"All the other players in the team have been vaccinated or have recovered – so they are more or less safe, at least from a medical point of view. One is never quite sure.
"Now, unfortunately, Granit caught it. From a sporting point of view, it is a shame for us at this important moment."
Arnold said the Switzerland team were doing all they could to reduce contact within their ranks, with social distancing encouraged where possible and masks worn in meetings.
"I think it would have been irresponsible on our part not to know who in the team is not vaccinated," Arnold said.
"We had intensive discussions with the players already during the Euros. We have done everything to facilitate access to the vaccine, but each person is free to decide whether or not they want to be vaccinated."
It appears unlikely Xhaka will be able to play in Switzerland's next game, which comes against Euro 2020 winner Italy in Basle on Monday (AEST).
However, Arnold, quoted in Blick, said: "We still have hope. On Thursday he will do another PCR test. It is the case that [on Monday] he had a negative PCR test, on Wednesday morning a negative rapid test, now unfortunately a positive one. So we still have the hope that the test on Thursday will be negative."
Arsenal sit bottom of the Premier League table heading into the clash with Norwich City following the international break, though regardless of Xhaka's health by then, the midfielder is facing a three-match suspension so will not be involved.