Inspired by this question What is earliest version of a "horcrux"?; is the Ring keeping Sauron alive?
I always thought that the Ring was only a source of power; while without it Sauron would be too weak to influence anything, the survival of Sauron's spirit is not enhanced by it; he would have survived the Fall of Numenor simply by being a spirit (and Eru not willing to eliminate him directly)
I realised that the concept of death for an Ainu is ill-defined. I think that it's better, for the purposes of this question, to assume that the death of an Ainu means that the spirit is unable to influence the physical world. That is, the question can be rephrased to:
If Sauron never made the Ring, would he be able to manifest again and influence the events in Middle Earth after the Fall of Numenor (or after his defeat in the War of the Last Alliance)?