Summary: you break everything.
I've read that the laws of physics can be thought of as non-laws, and the lawlessness naturally gives rise to Newton's laws. That is, there is a lack of special cases and special things. What remains and is self-consistent?
Newton's first law as understood in modern formulations states that there is such a thing as an inertial reference frame where an undisturbed object remains in uniform linear motion. If the universe has no origin coordinate, no absolute positions are possible, as no law can be position dependant. Similarly, all directions are the same.
Relative only (not absolute) motion exists because there is no absolute reference.
In order to impose a rule as you ask, that refers to absolute rest, there must be an absolute coordinate system in space itself, a place that can be identified without regard to anything in space.
You would not have general relativity. So it must be like Galelleo's model and has absolute universal time, too.
The acceleration keeps adding energy, which grows without limit. Energy is not concerved. But isn't conservation of energy a consequence of physics working the same at any given time? We're not varying the rules, so the underlying principles of Noether's Theorm must not apply. That is, the laws cannot be formulated on a principle of least action. There is no such (useful) concept as potential energy, no Hamiltonian, and no way to express the rules using a stricly local infinitesimal patch of space around a particle. Laws must be interpreted using global knowledge of the state.
It would be, essentially, game pieces controled from without, not laws existing within the universe doing the interpretation.
It would not exist in the same sense as ours. It might be a simulation in a larger universe that does have self-contained rules.
As others have pointed out, everything would get faster and faster and no structures would form. But I point out that without special relativity and with separate space and time, it will behave classicly, not form black holes or approach asymptotes.
You would not get pair-production and annialation as a required outcome of symmetries, either. What stuff exists won't naturally have decay pathways since that happens because of vacuum pair production or annihilation. It won't have quantum spin, since that (one of the triumphs of physics) appears naturally in the equations of motion due to spacetime being one thing. That's what makes matter act like solid stuff. Why is a brick solid and not able to pass through other bricks? Not from any rules introduced for the purpose (lawlessness!) but as an inevitable consequence of a cascade of emergent properties as you work out the details of having no "special" rules.
Everything about that universe needs to be programmed directly, because it's not all snowballed from the most primitive symmetry of spacetime. Good luck getting it all to fit together without contradictions and bugs.
For more on the main breaking point, check out Feynman's Messenger Lectures. It's in book form, and the live presentation is on youtube! He talks about different ways to formulate the same rule, and the 3rd way introduces "least action". That turns out to be the keys to the kingdom.
Not having conservation on energy implies that the "action princple" is not a way to formulate the laws in your proposed universe.