PC Ian Bell, 46, admits sending the messages while off-duty and drunk. They include one which said 'You like sex?' and another which read 'Please delete everything'.
He accepts his actions amounted to misconduct but denies gross misconduct, a sackable offence.
The girl was living in a children's home and PC Bell had been investigating reports that another young resident had gone missing, a misconduct hearing in Wakefield was told today.
He visited the home on July 19 last year and had a conversation with the girl, who cannot be named.
In the early hours of the next day, he began sending her messages via the social media application Snapchat which had "no legitimate policing purpose", Claire Watson, legal representative of the force's Professional Standards Directorate, said.
She said: "He knew, or ought to have known, that his actions were an abuse of his position as a serving police officer and was likely to cause distress and anxiety to the young person to whom he was sending these messages."
At first, the girl told the children's home staff she thought she had a "Snapchat stalker" and did not know who was sending the messages, the hearing was told.
Ms Watson said that at about 1am the girl then asked a member of staff: "If it was a police officer who sent the messages, what would happen if he could come and find me?"
She said: "The messages included references to seeing [the girl] in a towel, a question about whether she was taking drugs at parties and whether she likes sex."
Adam Birkby, representing PC Bell, said: "This is a case of drunken idiocy and not a sophisticated attempt to exploit a vulnerable person for sexual purposes."
The misconduct hearing was told that PC Bell sent a message telling the girl he had seen her wrapped in a towel while he was at the children's home.
He asked the girl why she was not in bed "in your PJs" and she said if she was tired the next day she would not go to school.
He asked her whether she took drugs such as cocaine.
He told her not to let anyone take advantage of her, and she said: "I don't care if they do, TBH."
He replied by calling her "cheeky", then asked if she liked sex.
PC Bell has given 23 years of service to West Yorkshire Police and was based in the Wakefield district, the hearing was told.
Giving evidence, PC Bell said he rated his drunkenness as eight out of ten on the night in question and started sending the messages to the girl after his wife went to bed.
He said he got the girl's Snapchat details from her Facebook profile and wanted to thank her for her help in the missing person's enquiry, as well as giving her advice.
He said: "She seemed quite a bright individual, vulnerable in the fact of leaving herself open on Facebook, but she could be diverted away from getting herself into any more bother.
"You could just tell, from speaking to her, that there was an opportunity that she could be saved from going into crime and diverted away."
PC Bell said he was concerned the girl could be taken advantage of but accepted that this had been an inappropriate way of acting on his concerns.
He said he shut down the conversation and deleted Snapchat from his phone because he was "panicked" at how the conversation could be misconstrued.
He said: "I was off-duty, intoxicated, talking to a 14-year-old person. I let the force down and I let myself down."
Cross-examining PC Bell, Ms Watson disputed his claim that he had been wanting to prevent her from being taken advantage of.
She said: "In reality, is that not exactly what you were doing: contacting her and grooming her by escalating the suggestive nature of the messages you were sending?"
"No," he replied.
Ms Watson said: "It was for no other reason, PC Bell, than for your own sexual gratification."
He said this was not true, adding: "I haven't done it with the intention of a sexual motive. My wording could have been a lot better."
The misconduct panel will decide tomorrow whether PC Bell should be dismissed.