Can my employer ask for a specific diagnosis, if a mental disability has already been stated by a doctor in writing?

Yes, an employer may, for three reasons:

1) First, laws applying to not discriminating against the disabled only apply to mental illnesses that qualify as a disability--that is, are recognized sufficiently by the medical profession, are not easily or readily remediable or not temporary, etc. Since if something is a true disability, it will affect the employer's legal rights under anti-discrimination law, it has the right to get the exact diagnosis and make sure this is a disability.

Related to the above, the accomodations an employer may have to make are related to what are required by *this* specific condition, if it is a disability; this is another reason why it needs more data.

2) Similar to the above, the employer's potential coverage and exposure under different insurance (including health insurance and liabiltiy insurance) laws may be affected by the the nature of the disability; the employer has a right to this information to make sure it is complying with its policies and getting all the coverage due to it.

3) Some mental disabilities can result in the person with the disability posing a threat to himself/herself  or others. An employer is not required to blindly accept this risk, but may ask for information to understand what the risks are. If you do pose a risk to others and the employer does not follow up to understand those risks and take appropriate action, it could find itself liable for significant damages if anything happens.

If you had a "heart condition," the employer could seek claritification as to what it was and how it affected your work and the risk you'd be injured at work; if you had a "back problem," it could ask for more data to understand what you can and cannot safely do; the fact that this is a mental disability does not change the employer's legitimate right to more  information.