Am I obligated to offer expiation or repentance for accusing a veiled girl, specifically one wearing a "loose niqab," of being a prostitute, knowing that I know nothing about her, and that my accusation was based on my observation of her laughing and talking with a young man in the street, a behavior that might arouse suspicion in my country?

You must repent to Allah for delving into the woman's honor. Repentance is achieved by abandoning the sin, regretting it, and resolving not to return to it. It is not necessary to seek the woman's forgiveness due to the impossibility of identifying her or the fear of greater harm. Repentance between the servant and his Lord suffices. There is no harm in seeking forgiveness for the woman and supplicating for her in her absence.