Is it permissible to give money to the institution's doctor to obtain sick leave?

Bribery is forbidden and is considered a major sin. It is not permissible except in cases of dire necessity, such as to obtain a right or to repel injustice. If the rest advised by the other doctor is not necessary and does not entail any benefit, then it is not permissible to pay a bribe. However, if the rest is required, and neglecting it would cause harm, and the doctor's refusal is an obstinacy to force you into bribery, then there is no objection to paying it to obtain the right and repel harm.