PC Ian Bell, 46, who had been based in the Wakefield district, was dismissed without notice on Tuesday after a two-day misconduct hearing held in Wakefield.
The hearing was told that PC Bell had been arrested over the incident, but a spokeswoman for West Yorkshire Police has since said that he will not face charges.
She said the Crown Prosecution Service had “considered the criminal case and declined to prosecute”.
PC Bell had met the teenager on July 19 last year while investigating the disappearance of another young resident of the children’s home, the hearing was told. The next day, while off-duty and drunk, he looked up her contact details for the social media application Snapchat and sent her messages.
They included references to having seen her dressed only in a towel at the home, asking whether she took drugs and whether she liked sex.