Yes it would, ...probably.
A lich Rejuvenation ability is triggered when the lich is destroyed. The petrification condition states that a petrified creature is incapacitated and transformed to stone, but not destroyed. So, if you manage to petrify a lich, it will become a statue and until it is de-stoned... or someone smashes it into pieces. A petrified lich is a really resilient statue, retaining all the lich damage inmunities and hit points and gaining resistance to all damage to boot, but it can still be destroyed with magic weapons, spells, acid, etc. All it takes is one or more well prepared minions infiltrating your garden at nigth to destroy the lich, enabling it to use its trademark escape clause and return later to enact a carefully planned revenge. Be sure to keep the statue well-watched and take advantage of the lich state to find and destroy its phylactery.
Also, if the lich happens to have the phylactery on its person when petrified (not very probable), the item will become lodged into the stone (being a magic item spares it from petrification) and you'll have an interesting choice: Leave it as it is, and hope that nobody breaks the statue, or try to extract the phylactery from the statue without breaking it yourself. If you choose the later and worst come to worst, you will have 1D10 days to find a way to destroy the phylactery (not an easy task) or to prepare for a rematch with a angry lich after its new body reforms next to the phylactery.
Let's consider a different question for a moment. Suppose I'm building a monster, and I give it a melee greatsword attack with +4 to hit and 2d6+4 damage. How much CR does that add?
Well, it depends. If my monster was a huge dragon and it already had a bite attack that's way better than the greatsword, then the greatsword never gets used and it adds no CR at all.
On the other hand, if my monster was a ratling and its previous best attack was a dagger for 1d4+1, then the greatsword is actually a big deal.
So the answer I'd propose is: if you add an attack which the monster is going to use as its primary attack, you don't "add" to the Offensive Challenge for that. What you do is you figure out the Offensive Challenge for that attack, and you have just set the monster's Offensive Challenge to that number.
So: what is the Offensive Challenge of a petrification attack? What other monsters can we find that have petrification attacks, and what is their Offensive Challenge?
The basilisk has a similar power, except it's attached to a Constitution saving throw, DC 12. This power is the basilisk's most dangerous attack, so the Offensive Challenge of the attack should be about the same as the CR of the basilisk. The basilisk is CR 3, which, uh, seems kind of low actually for a save-or-die attack. The CR for your power might be lower still, since it requires an action to use and only affects one target at a time.
The medusa also has this power, except in the medusa's case the power is DC 14 and the CR is 6. Maybe we could make sense of this by saying that adding +2 to the DC means +2 to the CR, and the medusa also has a snake hair attack which is pretty dangerous, so maybe that increases the CR by 1 since the medusa can do both at once.
So I think the answer to your question will depend on the DC of the ability.
If I could add a note, as well: in my experience, fighting monsters with save-or-die attacks is frequently not fun. Either the player character makes the save and the monster's attack does nothing, or the player character fails the save and now that player is out of the game. You might consider replacing the save-or-die with something that does (for example) ability damage.
Best Answer
The Petrified Condition would not be suppressed or removed unless it is magical
Antimagic field has a specific list of effects stating what it is capable of doing - and the one most applicable to this situation would be:
Petrified is neither a Spell effect or a Magic effect, but a condition, similar to being grappled or exhausted.
If you wanted to remove petrification, you need to use a spell that specifically states it is capable of doing so. For example, the spell Greater Restoration:
However, when the effect is applied by a Beholders eye rays, it is considered magical:
Therefore - magical petrification would be unable to affect a target within the sphere, and both the on-going saving throws & petrification condition itself would be supressed if the targets enter it while affected.
Does suppressing petrification stop the on-going petrification effect?
If the petrification is considered magical, it will be suppressed, both by Antimagic field & the Beholders own Anti-magic eye cone.
If the effect is suppressed, the target doesn't make a saving throw, but it also states that the effect is only ended on a success. Therefore the effect continues until the target succeeds on a saving throw.