The Rutterkin (from Mordenkainen's Tome of Foes, pg. 136) has the following attack:

Bite. Melee Weapon Attack: + 4 to hit, reach 5 ft., one target.
Hit: 12 (3d6 + 2) piercing damage. If the target is a creature,
it must succeed on a DC 13 Constitution saving throw against
disease or become poisoned. At the end of each long rest, the
poisoned target can repeat the saving throw, ending the effect
on itself on a success. If the target is reduced to 0 hit points
while poisoned in this way, it dies and instantly transforms into
a living abyssal wretch. The transformation of the body can be
undone only by a wish spell.

(Bold emphasis mine)

A level 3 Paladin has the Divine Health class feature (PHB, pg. 85):

By 3rd level, the divine magic flowing through you makes you immune to disease.

Recently, our party's level 3 Paladin got bitten by a Rutterkin and failed the Constitution saving throw. The DM has ruled that he is now poisoned (although hasn't told the player yet). The player (correctly) suspected that it might be something disease related, but the DM says it still counts (he has explained to me privately that it's because it gives the poisoned condition, not a disease).

Would a Paladin with Divine Health be able to automatically succeed the saving throw against a rutterkin's bite?

On the one hand, the saving throw is "against disease", but as my DM points out, the effect is gives is the poisoned condition rather than a disease specifically. Which interpretation is correct?

The Paladin is immune

The fact that the saving throw is specifically against disease, and that Paladins are immune to disease means they should not have to make a save against this effect, regardless of what condition it applies on failure.

I suspect that the 'poisoned' condition is a way of representing the penalty of being diseased since there isn't a separate 'diseased' condition, although this is 'designer intent' territory. (It would also mean that a character immune to the 'poisoned' condition would also not be affected regardless of their save - technically they may have the disease but are just immune to its effects).

It cannot poison the paladin.

A disease, mechanically, is not a condition. It itself can impose conditions to PCs or provide other effects. The paladin is immune to disease and hence immune to its conditions and effects that he would otherwise suffer from (in this case being poisoned and turning into an abyssal wretch).

If a character was immune to being poisoned (instead of being immune to disease), it would also be fine (since that is the main effect of the disease).