The bus was shot at Saturday near the northern city of Trabzon on the team's return to the airport from a 5-1 win against Black Sea side Caykur Rizespor. The driver was hit in the face and hospitalized. He is in a stable condition.
Fenerbahce officials said the attack was carried out with a shotgun.
The Trabzon chief prosecutor's office said Monday it has launched an attempted murder investigation and that one man had been detained for questioning for making threats against Fenerbahce over the Internet.
The Turkish Football Federation and the Union of Turkish Clubs announced after an emergency meeting that they had decided to postpone all games by a week in protest of the attack.
Federation head Yildirim Demiroren said, however, that they had ruled against a cancellation of all remaining league games to avoid "rewarding the perpetrators."
"We believe that the incident is a terrorist act," Demiroren said. "Secondly, we took this decision because we attach importance to the players' psychology."
Goksel Demirdag, who heads the Turkish club's union, said: "This was an attack on Turkish sports. It was an act against all colours."
Fenerbahce vice-president Deniz Tolga Aytore told reporters Monday that the attack was an "assassination attempt" and a "terror" act that could have killed 41 players and club officials inside the bus.
"We have no intention of playing until light is shed on the incident," Aytore said. "To play ball isn't all that important when the lives (of players) are concerned."