Bedoya became the third American player – Carlos Bocanegra and Maurice Edu are the others – on the books at the champions when he signed from Swedish club Orebro this month.
Despite the fact he was nursing a heavy cold, the 24-year-old winger made his debut for the Gers when he came on as a substitute in the second half of an unconvincing 2-0 triumph over Aberdeen on Sunday.
Now he is itching to play again and prove his worth to Ally McCoist's team by helping them perform well in the three domestic competitions they are still involved in this season.
The 13-times capped player is not involved in his national team's friendlies with Costa Rica in California on Friday night and Belgium in Brussels next Tuesday evening.
USA coach and Germany legend Jurgen Klinsmann wants him to concentrate on adjusting to life in Scotland instead after finally completing his protracted transfer last week.
But Bedoya is keen to play a part in his country's matches alongside Bocanegra and Edu, who are both away with the Stars and Stripes squad at the moment, in the future.
Juggling international commitments on the other side of the Atlantic with club football for Rangers will prove hugely problematic for him, as it has done for his compatriots in the past.
However, he has no concerns that having so many United States players will have an adverse affect on his new club's bid to land a fourth consecutive SPL title.
He said: "I don't think I'll have any problems with playing for Rangers and playing for the United States. Sure, it can be tough sometimes being an American player based in Europe. There is a lot of travelling involved.
"For instance, our game with Costa Rica this week is in California on the west coast. Then the squad will go from there all the way to Belgium. There is a lot involved in fulfilling those two fixtures. It is just part and parcel of playing for the United States.
"But I don't think it will be as difficult in the future as there is going to be a change in the scheduling of our fixtures to make it easier for the European players to get involved. More games will be held on the east coast, in New York say, or in Europe to make it easier for us.
"But I am well used to it by now. I have been playing in Sweden for the last couple of years and I have travelled back on numerous occasions just to play in midweek friendlies. You travel on Monday, play on Wednesday and then head back again on Thursday.
"I think when you play for the United States you get used to it. I have no doubt that Maurice and Carlos are both quite comfortable with the situation as well. It is not as if we will be away every other week anyway. Plus, we will have no European football this season."
Bedoya, who became a Gers player at long last when Orebro agreed to release him for a nominal fee before his contract expired in January, is grateful to Klinsmann for the chance to settle in to his new surroundings.
He admitted: "Life has been pretty hectic for me to be honest with you. I have been packing up my apartment in Sweden and trying to get across to Scotland as soon as possible.
"While all that was going on, too, we were trying to get my visa sorted out. It all took longer than we had estimated it would. I was delighted when we finally got it sorted out last week.
"I have been glad to get to Rangers and start training with all my new team-mates. I'm excited about what is a great opportunity for me and I am looking forward to really getting going after the international break."
Bedoya added: "I had been under the weather for a week so it was difficult to make an impression when I came on. That could have been down to a change in weather or the travelling I have been doing.
"I think having some time to get myself fully fit and settle in to life in Glasgow will be a good thing for me. Hopefully, I will be in better shape when we play Dundee United in the league on Saturday week.
"Not having European football is disappointing, but we have to concentrate on the league."