It's possible to learn just the lead but I think that you will be putting yourself in a bad position as a player of music in general. Depending on your aspirations, playing chords may be necessary, as others have mentioned, such as playing in a band with only one chordal instrument or what you'd be doing during the singing.
Even if you don't intend to play in bands or anticipate playing chords in any setting, learning chords will help you a whole lot. Firstly, the coordination it takes to correctly place your fingers and transition between chords carries over into your lead playing. The shapes of the notes in different chords may end up being the same fingerings that you would use while playing those notes individual in a lead part and having learned and ingrained the chord shapes, playing that line will be much easier. You'll also find that lots of lead lines actually include chords. Usually the chords would end up having fewer notes (not using all 6 strings), which is also the case with lots of chords a rhythm player would play, such as in reggae or jazz, and some basic chords don't include all the strings anyway, like a D Major in open position.
If you were to pursue playing monophonically (one note at a time), then I would suggest looking at and trying to learn/emulate other monophonic instruments, such as woodwinds or horns. Strings would also be good, as they are usually playing monophonically but also have the capability of playing a couple notes at once. Even if you end up playing chords, I'd still recommend doing this since most instruments have a few different tendencies and don't necessarily overlap with other instruments very commonly, so learning saxophone solos might diversify your palette and bring a unique style to your playing.