Learn Hardcore Russian: Врач ставит печать на документ. - The doctor places a seal on the document.

The verb ставить is imperfective, meaning an ongoing or habitual action (“he is putting/places”). In this collocation it describes the action of affixing a seal. Its perfective counterpart is поставить.

Imperfective (process/habit): врач ставит печать
Perfective (single completed action): врач поставит печать (future) or врач поставил печать (past)

Why is печать in the nominative case and документ in the accusative with на?

печать is the direct object of ставить (“to put what?”). In Russian, direct objects of transitive verbs take the accusative, but since печать is feminine and ends in a soft consonant, its nominative and accusative forms coincide.
на документ uses the preposition на plus accusative to indicate direction onto a surface (“onto the document”). So the doctor places the seal onto the document’s surface.

What does печать mean here, and how is it different from штамп?

Печать is an official seal, often embossed or metal, used to certify documents.
Штамп is a rubber stamp, usually for office stamps with ink.
You’d use печать when referring to a formal, legally binding seal.