From memory — can't find the thing right now! — this would be a little difficult on the one I had because of its shape.

Also because — not sure how to explain it — but in my experience the holes on a four hole ocarina are also sort of finger grips, in a way. That is to say, you support the ocarina against your lips with at least one finger below it and two thumbs behind it, but part of your grip on the instrument is gently transferring from fingertip to fingertip as you play, a bit like a recorder or the pipe of bagpipes. The "open" low note has the loosest grip, and you might even subconsciously tilt your head back slightly to allow the weight of the ocarina to shift more to your thumbs.

It might be possible to design, effectively, a one-handed playable instrument that works the way you are talking about, but I think it would be quite uncomfortable.