I agree with everyone here about being accountable with one's actions and apologizing only if you did something wrong. Again, if I person is unattracted to you, they might think you are trying to gather unwarranted sympathy by saying sorry and they might find you pathetic. However, if they are attracted to you they my think is cute, vulnerable, etc.
However, in a grand scheme of things, there is a reason why politicians never apologize. Apologies are often seen as admissions of guilt and can lead to more criticism for public figures, as they confirm the original complaints. Supporters might also feel betrayed, making the apology counterproductive. Overall, apologizing rarely improves one's public image.