An implication exists for some clerics, of certain deities. There also have existed, for multiple editions, clerics of certain deities for which no such implication exists; or even, indeed, the opposite implication. There are other classes for which similar implications exist for specific builds. It is for this reason I stress the difference between a "cleric" and a "healer".
If enough pressure exists that the player in question feels the need to spark a public debate about whether it is the DMG's responsibility to address his plight; I am comfortable quantifying it as bullying.
I do recal playing a specialty priest in AD&D 2e who couldn't heal. He could inflict wounds, but he couldn't cure them.
Of course, the party only needed a healer if the DM failed to provide an alternative method of healing via potions, magical items, hirelings, or advancing the time. I'd say that bullying a player to be "the healer" is a poor attitude to have, and it's against the spirit of the game, especially when there are other options.
Non-magical healing is a modern concept that wasn't needed in the past. I understand that some people really like that idea, but it's just not my preference.