This is a great question! I believe the answer is no, but of course rule of fun, D&D is what you make it, blah blah blah.
Here is the PHB, Chapter 10 on "What is a Spell":
A spell is a discrete magical effect, a single shaping of the magical
energies that suffuse the multiverse into a specific, limited
expression. In casting a spell, a character carefully plucks at the
invisible strands of raw magic suffusing the world, pins them in place
in a particular pattern, sets them vibrating in a specific way, and
then releases them to unleash the desired effect — in most cases, all
in the span of seconds.
Spells can be versatile tools, weapons, or protective wards. They can
deal damage or undo it, impose or remove conditions, drain life energy
away, and restore life to the dead.
Uncounted thousands of spells have been created over the course of the
multiverse's history, and many of them are long forgotten. Some might
yet lie recorded in crumbling spellbooks hidden in ancient ruins or
trapped in the minds of dead gods. Or they might someday be reinvented
by a character who has amassed enough power and wisdom to do so.
Well hey, look at that! Spells restore life, so we're good right? Not necessarily. Cure Wounds is certainly a spell that restores life, but that doesn't mean that all abilities that restore life are spells. For example, a healer's kit restores life, but is certainly not a spell. Then again, unlike a healer's kit Lay on Hands is definitely magical: "Your blessed touch can heal wounds," says the PHB.
The real damning text is back in the Chapter 10 definition of spell, which states "Every spell has a level from 0 to 9." Lay on Hands improves as you level up, but there is nothing suggesting that this feature has a "spell level". The chapter continues to explain that spells are associated with spell slots, suggesting that anything that does not require a spell slot to cast is not a spell.
Lay on Hands is not a spell; it is a "Class Feature". It is listed under "Features" for the Paladin class description.