Would putting money in a stock with a beta of 1 be the same as investing in an index fund?

Beta is just a measure of past correlation. it is not indicative of what the stock might do in the future.

theoretically, that is what should happen.

I think beta is calculated within a sector. So if beta stays at 1 then its equivalent to investing in the sector index fund if such a thing exist (not in nepal ofcourse). But mostly index fund i see are a best performing stocks in a exchange that are mix of different sectors.