The real issue is that this (like all the other Scarlatti sonatas) wasn't written for piano.
A superficial argument is "harpsichords don't have sustain pedals, therefore don't use any piano pedal". But that ignores the fact that harpsichord playing technique is quite different from piano.
To get the maximum resonance from the instrument when playing a harpsichord, in general you hold down notes for as long as possible when they form the successive notes of chords, as in most of the right hand of this sonata. A harpsichordist would actually play the first two bars something like this:
The first 1:30 of this demonstrates the general effect, and shows that the harpsichord really can sustain notes if you want to play it that way.
You can't imitate that literally on a piano, because the piano action feels quite different from a harpsichord (in particular, it takes more effort to hold the piano keys down after playing the note) and on the piano, unlike the harpsichord, you have to control the dynamics of each note as you play it. Also, the plucking action of the harpsichord gives each note a better defined start point (more comparable to the start of a guitar note than a piano note), and the harpsichord tone decays faster than a piano.
So you have to use the piano sustain pedal to imitate this effect as best you can. Exactly how to do that depends on the particular piano, and the acoustics of the room you are playing in. The more resonant the piano and the room is, the less pedal you need to use.