F# has recently caught my eye as something I'd like to learn more about, but I have never used the .NET platform and know very little about it. If I want to write a small website in .NET core can I do that with only F# or will I need to expand my learnings to include C#?
This depends on your learning and working style.
Technically, F# does not need any support from C#. You can write complete programs of any kind whatsoever without ever touching C#.
However, a lot of F# resources online are geared towards migrating from C# to F#, and as a result are phrased in terms of the differences between the two. Something like "you've been doing this thing in C# before, but here's how it can be done way better in F#".
Additionally, a lot of .NET libraries are written in C#. The binaries are, of course, completely compatible with F#, so you don't need to do any sort of special tricks to use C#-written libraries. However, if you ever want to look at the source code, that's where C# knowledge would come in handy.
To be fair, there is quite a large number of F#-exclusive libraries out there, but the majority of .NET libraries are still in C#. So chances are, you will end up using some of them. Whether you would actually want to look at their source - that depends on your style.