A DM can certainly adapt to a party without healing, and there are so many other classes that grant healing, but the cleric offers a ton of utility/curative other than healing (revivify, lesser restoration, bless, etc.).
Like others have said, you can play without clerics but I would not eliminate them unless your campaign has an edge where gods are absent (like Dragonlance did).
I really like playing clerics, especially now that there are domains that dabble in combat or fire/spellcasting. I'm playing a Tempest Cleric in a campaign now and I love it. In some campaigns I've run over the past year, players have run War Clerics and Light Clerics. They've all had a blast.