When you write the roman numeral denomination of the chords, you are implying that the notes that make up each chord are present, although not explicitly written in the score. So "I" in your example means that the notes F-A-C would be in some fashion played in an improvised manner by a performer or arranged or orchestrated by the composer for the orchestra or choir, etc.
To answer your question literally, I suppose that yes, in some situations it may be possible (depending on what came before) to convey a clear harmonic function with just a two note melody. But it would be a very particular effect sought by the composer. And anyway, as explained, that's not what is intended in the example you provide.
Finally, about the A in the melody yes, in this case it is still under the F chord. This is called a "chord tone" (other than the tonic) as A is part of the F major chord. There could be other possible harmonizations for this note (.e.g a iii, or A minor chord), but the composer has decided that this note is still under the influence of the tonic, or F chord.