No, they can't. Since AIDS is not transmitted by airborne droplets. And dismissal from work affects the rights and freedoms of this citizen ... p
First, it is important to define the concepts. There is a concept of HIV infection - it is an infectious chronic disease that is caused by the human immunodeficiency virus. AIDS is the last stage of HIV infection, when a person has pronounced disorders in the activity of the immune system and the so-called AIDS-associated diseases occur. That is, a person may have HIV infection, but at the same time he may not have AIDS. These are not the same terms.
HIV testing is voluntary, anonymous and confidential. Potentially, the employer may not know (and does not have to know) that a person has this disease. Everyone decides for himself whether to report this information to his employer or not.
An exception is the list of specialties when an HIV test is mandatory and this is stipulated by law, in particular by the Decree of the Government of the Russian Federation of September 4, 1995 N 877, in addition, similar documents are in the paramilitary departments - the Ministry of Internal Affairs, the Ministry of Emergencies, military. In all other cases, a person has the right to refuse to take the test without giving reasons.
Receiving a diagnosis of HIV infection by a person is not an automatic reason for dismissal. In some situations, HIV infection is a contraindication to work in a particular specialty. In such a situation, the employer should transfer the person to another position, if possible. The issue of aptitude is not decided by the employer, but by a special medical commission.
In life, employers can often demand tests where it is completely unnecessary and he does not have the right to do so. In such a situation, you can go to court to declare his actions illegal and protect your rights.