Does Allah's saying: "Allah will raise those who have believed among you and those who were given knowledge, by degrees" include modern sciences such as medicine and engineering?

The noble verse is in the context of showing proper etiquette in the gatherings of the Prophet, peace and blessings be upon him, and others. It is general in its wording and meaning, and the reason for its revelation does not restrict the general applicability. The verse includes all beneficial knowledge through which its possessor intends to draw closer to Allah and fulfill a communal obligation (fard al-kifaya).