RAW, paladins cannot cast ritual spells as ritual
In the Spellcasting chapter in the PHB, under Rituals:
To cast a spell as a ritual, a spellcaster must have a feature that grants the ability to do so. The cleric and the druid, for example, have such a feature. The caster must also have the spell prepared or on his or her list of spells known, unless the character's ritual feature specifies otherwise, as the wizard's does.
– Rituals, PHB, p. 202
As an example of a spellcasting feature that does mention it, we might as well look at the cleric's:
You can cast a cleric spell as a ritual if that spell has the ritual tag and you have the spell prepared.
– Spellcasting, PHB, p. 58
The paladin's spellcasting feature (pp. 84-85) does not mention this anywhere, yet it does explicitly describe many things that are also described in the cleric's spellcasting feature description (p. 58), such as how to prepare spells at the end of a long rest, what spellcasting focus you use, etc.
If this stuff was supposed to be "assumed" from the cleric's spellcasting feature, why spell it all out again? The "Spellcasting Focus" part in particular is worded exactly the same in both the cleric's and the paladin's spellcasting feature.
Also note that it would have to be spelled out again anyway if paladins were meant to be able to ritually cast their spell, because the cleric's version specifically mentions "cleric" spells, not "paladin" spells, so that would cause confusion ("but I have this cleric spell via the Magic Initiate feat, surely I can ritual cast it", etc).
The quote you have extracted, "learned to draw on divine magic and cast spells as a cleric does", I don't think so much importance should be placed on the exact wording of "as a cleric does", since they clearly do not cast spells like a cleric.
If anything, the ranger/druid comparison is actually more similar, since they at least both cast using Wisdom. Paladins, by contrast, cast using Charisma, not Wisdom like a cleric, so if anything, they actually cast spells less like a cleric than a ranger does compared to a druid.
The Ritual Caster feat can get around this to some extent
You mentioned a feat in your question; you probably already know what that feat is, but just in case, it's the Ritual Caster feat (PHB, p. 169). However, since you must choose the spell list of a full caster when you take this feat, you still wouldn't be able to ritually cast paladin spells, even if the spell has the ritual tag. I imagine in this case, it would make sense to choose the cleric spell list, since their spell list has a lot of crossover with the paladin spell list.