If you don't like the answer you got, you can always get a second opinion, of course.
I agree with Mikawa-san and btw Mike as well;
I dont want you to take this wrong but how can you be sure that you have strep throat?
He writes down "tonsillitis" in English on a piece of paper. He writes down "common cold" and says it would only spread if it was that.
The more you learn Japanese you will understand the word or phrase "kaze" which is a catch all phrase for a number of ailments.
I take it he gave you some medication as well right? If he did, I'll also bet that included with them was the anti-biotics that you were looking for in the first place. Remember to take them until they are finished, probably 3 to 5 days worth. Once you start on the regimine of taking the medication, even if you have strep throat, you will not be contagious after 1 to 3 days.
Also can you be sure you DONT have tonsillitis? Did you ever stop and think that MAYBE he knew what he was looking at when he checked your throat?
Are you aware of the differences in symptoms between strep throat and tonsillitis?
Does your throat pain not improve or lessen after drinking a sip of water?
Do you have any of the following symptoms as well, headache, high fever, stomach ache, vomiting, or severe tiredness?
Do you have any signs of a rash anywhere on your body?
If you have these symptoms you MAY have strep throat.
Or do these sound more like your symptoms;
Pain in the throat (sometimes severe) that may last more than 48 hours and be associated with difficulty in swallowing. The pain may spread to the ears.
The throat is reddened, the tonsils are swollen and may be coated or have white spots on them.
Possibly a high temperature.
Swollen lymph glands under the jaw and in the neck.
Headache.
Loss of voice or changes in the voice.
Only you know and like Mikawa san wrote you can always get another opinion. Why not just take the medication that the doctor gave you and let things go for a few days. Chances are you are going to get better and in the end it will be a learning experience for you.