If I hurt myself at work and got a doctor's note saying that I needed to be off for 3 days but my employer forced me to stay, what are my rights? 

Your doctor has no authority over your employer--the law does not require an employer to honor a doctor's note, as a general matter. Also, the law does not require employers to give employees time off from work for medical issue, unless either 1) the employer and employee are both covered by the Family and Medical Leave Act (FMLA) and the employee takes FMLA leave; or 2) the employer and employee are both covered under a state medical leave law.

To be covered by FMLA, the employer must have at least 50 employees working within a 75-mile radius; you must have worked there, more-or-less full time, for a year; and also, the condition must be sufficiently serious. Most state leave laws are fairly similar, so while you should check your state's law, it will likely have roughly comparable criteria.

If you are not covered by FMLA or its equivalent in your state (if any; some states don't have any leave laws), then the employer does not need to give you leave except as provided by its own internal policies. For example, if your company gives you paid sick days, you could presumably use them.

So if you are not covered by a leave law, and you are not also given time by the employer's polices (such as sick day policy), they don't have to let you take time off. If you did not show up for work in that circumstance, they could have fired you.