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.

The flesh to stone spell has instantaneous duration. Thus there is no “normal” aura after the spell is cast. But it is generally agreed (I can't find unambiguous RAW entry for that) that since magic is present at the time of casting, the spell leaves lingering aura as normal. The flesh to stone spell is level 6. It results in moderate aura at the time of casting and a lingering aura persisting for 1d6 minutes (see detect magic). After that no aura can be detected.

Your option after the time has passed is the true seeing spell.

The subject sees through normal and magical darkness, notices secret doors hidden by magic, sees the exact locations of creatures or objects under blur or displacement effects, sees invisible creatures or objects normally, sees through illusions, and sees the true form of polymorphed, changed, or transmuted things.

Other similar spells or spells specifically designed to detect creatures turned to stone may exist in other sources that I don't know. Alternatively you can work on researching such a spell with your DM if he/she allows spell research.

At the same time spells are not the only way to tell petrified creature from a statue.

In D&D 3.5 Spellcraft skill can be used to identify materials made by spells.

20 + spell level: Identify materials created or shaped by magic, such as noting that an iron wall is the result of a wall of iron spell. No action required. No retry.

Pathfinder has this option for Knowledge (Arcana) skill:

Identify materials manufactured by magic Arcana: 20 + spell level

Finally, you can undertake a regular investigation.

For example, you can learn what clothes the prince was wearing before disappearing, or you can find out what statues of said prince were ever made (if any) and where they are situated. Based on this knowledge you can make a decision on whether this statue is a regular statue or a petrified prince.

People in the comments also note that a statue will bear marks of the instruments used to sculpt it, thus a Profession (Sculptor) check may help to discover them.