Is it permissible for a man to post a picture of himself shirtless on social media knowing that girls may look at or with the intention that girls see the picture?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The ‘Awrah (parts of the body that must be covered in Islam) of a man is from the navel to the knee.

If you mean that that man published his picture while covering his 'Awrah, then he should not have published his picture on that condition even if he was covering his ‘Awrah, because this indicates a lack of shame and chivalry; by doing so, he may resemble (imitate) dissolute and immoral people who usually do this.

You will not almost find a bashful man of chivalry who would show himself to the people on that condition without a need in a way that he will be seen by everyone.

Ibn ‘Umar narrated that the Prophet said: "Whoever resembles a people, he is one of them." [Abu Daawood, and Ibn Hibbaan classified it as Saheeh (authentic)]

Allah knows best.