Yes. By the HTTP protocol, clause 7.2.1:
“Any HTTP/1.1 message containing an entity-body SHOULD include a Content-Type header field defining the media type of that body. If and only if the media type is not given by a Content-Type field, the recipient MAY attempt to guess the media type via inspection of its content and/or the name extension(s) of the URI used to identify the resource.”
So yes, the response headers should contain
Content-Type header for any response data (called “entity-body” in the protocol, often “file” in common language). If it is omitted, the browser is allowed to make its own wild guesses on the type of data it got. In many contexts, the risk of wrong guesses is negligible, but this not a good excuse for violating the protocol.
Content-Type, browsers will treat the data as CSS if
<link rel=stylesheet ...> was the element that caused the request. It’s the
rel attribute that matters in that case.