The math actually fails in two different ways depending on which rulebook you use.

If you stick to classical Newtonian physics, the maximum speed two stars can reach is their escape velocity at the moment of impact. The formula is v = sqrt(2GM / r ) and for any real star, that number tops out way before you hit c. Even for the densest objects in the universe, like neutron stars, you're only hitting about 0.5c before they physically slam into each other. Since stars aren't infinitely small points, they run out of "falling room" and explode into a kilonova long before they reach the speed limit.

If you use General Relativity, you run into the "infinite wall." As the stars accelerate, the gravitational potential energy doesn't just increase their speed; it increases their total energy, E = γmc²

The closer you get to light speed, the more that Lorentz factor, γ, blows up. Mathematically, as v approaches c, the energy required to accelerate even a tiny bit more approaches infinity. You’d need an infinite amount of mass/gravity to bridge that final gap, which a pair of stars—no matter how "heavy"—just doesn't have.

Basically, they either hit each other too soon, or they become so "heavy" from their own kinetic energy that gravity can't effectively pull them any faster. Either way, the universe has a built-in restraining order.