By Newton's Law of Universal Gravitation, the gravitational force between two particles is $Gm_1m_2/r^2$. Let's assume that the numerator is constant and happens to equal $1$.
Imagine that two particles that meet the above assumption are near each other in a vacuum. As they are colliding, the distance between them will at some point equal a Planck length. The force between them will then be: $1 /( 1.6 \times 10^{-35})^2\ \text{N}$. This is about $4\times 10^{69}\ \text{N}$.
Why doesn't this really happen?
Answer
I think the only issue with your scenario is using Newton's law of gravity to calculate the value of the attractive force between the particles. There is no reason to think that this situation 'cannot' happen.
In the situation you've described, each particle has a mass of $122,406 \text{ }\mathrm{kg}$, which yields a Schwarzschild radius of $1.818\times10^{-22} \text{ }\mathrm{m}$. But the Plank length is considerably smaller, at $1.616\times10^{-35} \text{ }\mathrm{m}$. To be within one Plank length of each other, they would both need to be black holes, with each singularity inside the other's event horizon (in addition to its own, of course). In other words, this scenario involves two black holes merging together. We seem to be well beyond the range of validity for Newtonian gravity.
Can two singularities get this close to one another? I guess so. Interpreted literally, a 'singularity' in GR is a true point particle.
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