Is it obligatory to pay a sum of money to a doctor in exchange for a certificate of exemption from physical education, especially since this sum was agreed upon after the doctor had initially apologized for providing the certificate, then agreed in exchange for money, and the certificate was received with a promise to pay later?

Bribery is forbidden and is considered a major sin. It is unlawful to solicit, offer, or accept it. However, the majority of scholars permit it if one cannot obtain their right or repel an injustice except through it, in which case the sin falls upon the recipient of the bribe. Based on this, paying money to obtain an exemption certificate is permissible for you if you are rightfully entitled to it for free. But since you obtained it without paying, you are not obligated to fulfill your promise, rather you should not; because it is unlawful money for the recipient. And if he demands it from you, inform him that it is not permissible for him, as Sheikh Ibn Baz, may Allah have mercy on him, clarified regarding unlawful debts.