The attack, involving a 17-year-old girl, happened at about 4.15pm on Friday as the bus was travelling from Ousegate to Foxwood Lane, Acomb.
Police said she was touched inappropriately by a man, who also made racially-abusive comments towards her.
Officers are appealing for witnesses to the attack and enquiries are ongoing.
A spokesman for North Yorkshire Police said: "We are requesting the public’s assistance to help establish the full circumstances surrounding the incident. In particular, we are appealing for information about from anyone who may have witnessed the assault take place or knows who is responsible."
Ayone with information is asked to call police on 101, quoting reference number 12180205631.