Divine Smite can't be stored in a Ring of Spell Storing.
Any creature can Cast a Spell of 1st through 5th level into the ring by touching the ring as the spell is cast. The spell has no effect, other than to be stored in the ring. If the ring can't hold the spell, the spell is expended without effect. The level of the slot used to cast the spell determines how much space it uses.
Divine Smite is not a spell of between 1st and 5th level. Since it's not, it cannot be stored into the Ring of Spell Storing.
You did not ask for the reverse, but likewise:
You cannot activate Divine Smite using spells stored in a Ring of Spell Storing
Divine Smite specifically calls out "spell slot" as the resource it consumes to be activated. However, the Ring of Spell Storing stores spells, not spell slots. The difference being that the spell is effectively already formed and cast whereas a spell slot is an untapped source of magical energy. So even if there are 5 spell slots worth of level 1 spells stored in the ring, you cannot spend any of them to use the Divine Smite ability.
What it means to be a Spell in 5e
While this is starting get get a little off-scope in my opinion, it may not be clear as to why Divine Smite is not a spell, even though it is powered by a spell slot.
This is an excerpt from the Player's Handbook Chapter 10 on Spellcasting.
Every spell has a level from 0 to 9. A spell's level is a general indicator of how powerful it is, with the lowly (but still impressive) magic missile at 1st level and the earth-shaking wish at 9th.
If it doesn't have a spell level, it's not a spell. End of story, right?
But what if you consider the Divine Smite's choice of spell slot to be the "level" for Divine Smite? I mean in that same chapter, there's this excerpt.
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting.
If we used a 2nd level spell slot to power Divine Smite, that would make it second level, because it assumes that level, right?
Still no. The reading of the first excerpt should be clear on it's own at this point: if it doesn't have a level to begin with, it's not a spell. Since it's not a spell, rules that change the effective level of spells don't affect it. There's no loophole here.
Almost always, race and class features are not spells. If there is an exception, I don't know if it. Whenever they involve spells, such as Tiefling's Infernal Legacy or Warlock's many invocations, they'll explicitly use the word cast, state which already-existing spells they replicate, and whether they are cast at different levels than their base. These types of features grant the ability to cast a spell, but are not a spell in-and-of themselves.
Divine Smite does none of these things, does not have an indicated spell level, and is not included in the list of Spell Descriptions at the end of Chapter 11 in the Player's Handbook (the book Divine Smite was introduced in). So, nothing about this feature involves an actual spell, just a spell slot.
The Paladin class does however have a wide variety of spells that function like Divine Smite, but require a higher action economy (an extra bonus action vs. Divine Smite's actionless trigger), occupy the Paladin's Concentration, and have weaker damage overall.