What is the ruling on a Muslim female doctor examining male patients in a clinic in North America, knowing that she is a trainee and needs this experience for specialization, and the country's laws do not permit the presence of a third party during the examination except for the patient's family?

It is not permissible for a woman to treat men or train them except out of necessity or need. If such is not available, then there is no harm, provided that Islamic precautions are taken, such as avoiding khalwa (private seclusion), not touching or looking except out of necessity, and wearing gloves, because necessity is determined by its extent.