Sound like you need a consult with an employment/labor lawyer who can draw the fine lines between your rights to privacy vs. and employer making sure an employee is not taking advantage by making up some medical excuse. If they terminate you (you should never resign) then your case might change to other discrimination type laws as well as medical leave laws (Federal and state laws apply). Also while your claim goes forward you can seek unemployment benefits to help your cash flow.
My answer is not intended to be giving legal advice and this topic can be a complex area where the advice of a licensed attorney in your State should be obtained.
While you do have privacy rights, the company has the right to have some information to determine IF you are able to work, what, if any restrictions your doctor has placed you on, etc. so that they can make sure that you are not a danger to yourself or others - they don't want to get sued. The Request for Medical Information form should not delve into too much detail. If you feel it violates too much of your privacy, retain an employment law attorney to help you complete it - the cost should not be very much. If you don't cooperate in a reasonable manner with your employer, they have the right to fire you. If you are out on FMLA or CFRA, your job is only protected for 12 weeks. If you are not eligible for FMLA and your employer has 5 or more employees, they have a duty to "reasonably accommodate" your medical condition - up to the point where it causes an undue hardship on the company. But, they need some basic info from your doctor (lifting, number of hours, etc.) to determine how to accommodate your med. condition.