If two stars collide, what is the probability that they merge to form a single star? After looking at What are the odds that the Sun hits another star? and answering it (crudely), now I'd like to ask the following:
What is the probability that if two stars collide, their cores merge to form one larger, more massive star?
Fairly good.
Two stars of mass $M$ falling from infinity straight towards each other until they merge at distance $2R$ will get kinetic energy $GM^2/R$. This is a lot, for two suns it is $1.8978\times 10^{41}$ J. However, compared to the binding energy of even a single star, $\approx 3GM^2/5R$ this is less(the sun has binding energy $2.2774\times 10^{41}$ J, and a double mass $2^{1/3}R$ radius same density merged star $7.2302\times 10^{41}$ J, 3.17 times more). So there is not enough energy released to blow up a star, but is is about a quarter of it: a lot of matter is going to get ejected or end up in orbits through a heated envelope that will take a while to simmer down.
The key issue is whether the cores get slowed down enough by the encounter to remain bound, becoming a common envelope binary. A direct hit clearly would work, but glancing collisions may allow the cores to miss each other: now the question is whether the envelope can absorb enough kinetic energy. A rough estimate may be that there is significant slowing if the mass scooped up/pushed aside $\pi r_{core}^2 \rho_{envelope} R$ becomes comparable to $m_{core}$. For two sun-like stars with $r_{core}=0.2R_\odot$ this seem to happen, but much hydrodynamics may occur complicating things.
Glebbek's dissertation on stellar mergers estimates a rough condition for the orbital angular momentum to exceed the maximum spin angular momentum of the merged star as $$\frac{r_p}{R_1+R_2} &gt; k^4\frac{(1+q)^{\xi+4}}{2q^2}$$ where $k^2\approx 0.05$, $\xi\approx 0.6$, $r_p$ the periastron distance, and $q=M_2/M_1$. This typically is exceeded: there is a lot of angular momentum that needs to be shed (for example by blowing off a lot of heated gas). For example, two sun-like stars having $r_p=R_\odot/2$ has a LHS of 1/4 and a RHS of 0.0303.
That dissertation also contains numerical simulations of various merger cases.