LGBT groups have repeatedly called on Bildt to condemn Russia's record on gay rights.
Bildt posted the message on the eve of the Stockholm Pride festival.
Separately, over the weekend, Bildt criticised Egyptian security forces in the wake of violent protests across the country.
On Twitter, Bildt said he was "horrified by the large number of dead in the demonstrations in Egypt." He added: "Security forces can't avoid responsibility."
The message provoked a response from Egypt's ambassador to Sweden, who said the protesters were in fact an armed mob.