It is a little bit hard to say whether there are inconsistencies or not. It depends on what you consider Newton's Universe to be.
For example, Newton invented calculus to help with calculations. Calculus involves infinitesimals. Infinity is filled with logical difficulties that are hard to put on a mathematically consistent footing. It wasn't done until around 1900. So maybe you could find inconsistencies in math as Newton did it. But it is possible to do that math right.
Newton's laws do not predict the world as it is. It is an approximation. So are the best theories we have today. So I understand that when Newtonian mechanics predicts that atoms are not stable, that isn't what you are talking about. But again, Newton didn't predict anything about atoms. The laws of electromagnetism came afterward.
So Newtonian laws are incomplete. They don't talk about E&M, strong, and weak forces. You can add them. If you do it naively, you get laws that don't describe the universe. If you fix the laws, you get away from Newtonian mechanics.
Newton did talk about point particles, forces and accelerations, and gravity. Also optics and other topics. If you start up the universe, mechanics will predict its future forever.
So you can talk about objects orbiting each other. But you have to invent what those objects are. You can't invent something realistic without relativity and quantum mechanics and future theories we haven't worked out yet.
So you invent something unreal. That might have inconsistencies, but that isn't Newton's fault. For example, massless frictionless pulleys are often used in high school physics lessons. What is the acceleration of such a pulley given a force of $0$? You get $a = F/m = 0/0$.
You can invent classical atoms where electrons spiral into the nucleus. That isn't real either. If there are logical inconsistencies with it, they are your own fault for inventing it. So infinitely dense point particles are not Newton's problem.
Gradients of fields are inconsistent at $r = 0$ for such particles. Fields were invented by Faraday, but you might consider that fixing up the math inherent in Newton. Sort of like fixing any problems with infinity. Or you might consider it a problem inherent in something that Newtonian mechanics can describe.
Some of the math that improved on Newton came from Leibniz. He also invented calculus, and not in terms of Newton's fluxions and such. More came from LaGrange and Hamilton. These rearranged Newton's laws into other useful forms.
The math of orbital mechanics was refined. When Ceres was discovered, only 3 observations were made before it disappeared behind the sun. It would come out in a few months, but nobody know exactly when or where. It was lost. Gauss took those few months to solve the equation. Along the way, he invented error analysis and the Gaussian distribution. Ceres was found within a degree of where he predicted.
All of this was both beyond what Newton did, and consequences of it. We haven't found any logical inconsistencies that were introduced.
One last point. At the end of the 1800's, physics was a solved problem except for a few loose ends. Relativity and quantum mechanics came as a surprise and overturned that. Likewise, Russel and Whitehead had put all of mathematics on a solid consistent footing. Godel's Incompleteness Theorem came out of the blue and showed that math is either inconsistent (unlikely) or incomplete (much more likely). We might find a logical inconsistency in Newtonian mechanics in the future. Like the correctness of physics theories, consistency can't be proven. Only disproven.