Can I get in trouble for going to a doctor’s appointment?

In an "at will" work relationship, a business can set the conditions of employment much as it sees fit. This includes terminating, suspending or otherwise disciplining a worker for taking time off to go to a doctor's appointment (this is true even if they have a note). An employee cannot miss work unless they have and use PTO to cover the absence; are following the company's policy about taking such time off (if there is one); and/or the company is covered by and the employee is eligible f FMLA leave under the circumstances. Otherwise, an employee has no inherent right to be absent from work other than for what might be provided for under the terms of an employment contract or union/collective bargaining agreement. Also, no form of legally actionable discrimination must be the reason for not allowing a worker to keep their doctor's appointment.

As a general matter, an employee has no inherent right to miss work for a doctor's appointment, no matter how much she needs it. An employee may be terminated for missing work for a medical appointment unless she has and uses paid time off (PTO) to cover the absence, or was eligible for and used FMLA leave (you can find the criteria for when you, your employer, and the condition/medical need are covered by FMLA on the U.S. Department of Labor website).
If you feel that you are being treated differently or worse than other employees due to your sex (e.g. that you are woman), your race, your religion, your age (if you are over 40), or disability, then you may have an employment discrimination case, and may wish to contact the federal EEOC or your state's equal/civil rights agency about possibly bringing a complaint. Otherwise, however, the employer may deny you permission to go to the appointment unless you can use and do use PTO or FMLA, and can terminate you if you go anyway.