No, "on" + date is fine:
1. On December 5th, I broke my ankle.
You use "on" + date to specify the day on which something happened.
You can also use "on" + a part of a day:
2. On the morning of January 17th, I broke my wrist.
3. On the evening of February 7th, I broke two ribs.
4. On the afternoon of March 20th, I broke three teeth.
Or even with a more general time-phrase:
5. On the following day, I threw away my lucky rabbit's foot.