Learn English – Is “on January first” is also correct

The first date is always more recent than the second

Sounds just fine, if you're sure it's true! (It is in this case, of course). In this type of situation, when you're looking for an adjective (comparative) to use, my advice is use the one most suited to the noun (thing) you're describing.

On face value, those are numbers in your example. But really, they're dates. Dates = time, so we need adjectives to refer to this concept of time.

Greater is for numbers (integers, fractions, etc).

You could explore avoiding an adjective altogether, for example:

The first date always comes before the second

Or, switching your order:

The second date always comes after the first. @JR

In American English, we don't normally write dates that way, but instead write and say with the month first — even if it's written day first, we'd still usually pronounce it month first. So "27 July" becomes "July twenty-seventh".

I believe other dialects usually, but not always, include the article before the day when putting the day before the month, so "[the] twenty-seventh [of] July". It will be understandable no matter which way you say it, but it's safer to use the slightly longer form to avoid sounding awkward in places that aren't that terse.