I wrote a (very) long reply to this digging into the motivations behind the development of GR which I didn't have a chance to post and which to some degree is made redundant by /u/jeffbell and /u/chunterbunter but I hope it may still interest you.
The short answer is "no, nowhere near", but let's dig into it properly. The TL;DR is that neither the maths, the observations, nor the technology to make these observations, were available to Newton - but that without him (and contemporaries such as particularly Leibniz) the later developments would have to wait until finally his results and techniques were found.
Opening statements
The first thing we have to do is to put the question in context. A "frame of reference" is how things look from a particular vantage point (such as sitting in a car on a motorway), and a theory of relavitity is simply a way of transferring how things look in one frame of reference to how they look in another. (I might use the words "inertial frame", too -- this is a non-accelerating frame of reference.)
The first important thing to note is when Newton was working, a theory of relativity already existed that lasted until the late 19th century: Galilean relativity. This is the relativity we all instinctively know. If you're driving at 70mph and someone passes you at 80mph, to you they look like they're traveling at 10mph. This is Galilean relativity, first properly formulated by Galileo, and assumed to hold in both inertial and non-inertial frames of reference. A few centuries later Einstein presented his theory of Special Relativity for reasons we'll examine shortly, and very carefully clarified that his new theory of relativity held only between inertial frames of reference. In a manner of speaking, one can view Special Relativity as generalising and extending Newton's laws of motion. While Special Relativity satisfactorily addressed its motivations and is one of the bedrocks of modern physics, it had (and has) a major drawback: it is impossible to incorporate a theory of gravity within its framework. And so Einstein introduced General Relativity, which is more properly a theory of gravitation replacing Newton's universal law of gravitation both in regions with extremely strong gravitational fields, and objects moving at high velocities.
We also need to put Newton in his context. Newton is primarily remembered in physics and maths for his groundbreaking work on optics, on the laws of motion, on the universal law of gravitation, and perhaps even more significantly the development of the differential calculus. Newton was working in the relatively early days of physics as a "modern" science -- based on high quality observations, such as those of Galileo for motion (and, indeed, the observation of the four main Jovian moons which helped lead to the acceptance of a heliocentric model), of people like Brahe for the movement of the planets, and his own observations of light based on the high-quality prisms and lenses now available. Galileo and Brahe's own observations (among of course many other astronomers) were only possible due to the increasing quality of lenses and mirrors. He was also relatively early in the development of modern mathematics; while maths was at a level of sophistication enough work with algebra and trigonometry and, therefore, predict the positions of planetary and satellite bodies, otherwise to modern eyes it looks extremely limited. To keep the example of planetary motion, the observations were more than precise enough to demonstrate gross mismatches with the theory, which posited that planets orbited the Sun in purely circular orbits, leading to increasingly convoluted networks of epicycles scarcely more satisfactory than the old geocentric models with their epicycles.
Perhaps the biggest missing tools were of course those of calculus, which Newton was instrumental in developing, along with in particular Leibniz. (Indeed, our modern notation is typically rather closer to Leibniz' than it is Newton's, for what that's worth. It does make reading the Principia rather trickier to read than I'd like. It's use of Latin to this non-Latin speaker also doesn't help...) But while by the end of Newton and Leibniz's times we had the foundations of calculus, it is extremely important to note that this was followed by centuries of solid development. This is particularly significant when considering the General Theory of Relativity.
So when we look into the theories of relativity and their motivations we must always bear in mind the observations available to Newton, the conclusions that he could possibly have drawn from those, and the tools available to him when attempting to model them.
(Continued...)
No, not really.
Newton was pretty well up to date on the experimental physics as it was known at the time. In Opticks on page 277 he mentions that Roemer had measured the speed of light by examining how the timing of the moons of Jupiter changes with distance. From this it was estimated that light takes seven or eight minutes to arrive from the sun, which is pretty close to modern measurements.
He also discussed how the speed of light will be faster or slower depending on the material, and applies this to estimating the amount of refraction.
Nobody anticipated the need for anything like relativity until more sensitive experimental results of the nineteenth century showed that the speed of light was the same for all observers. These measurements led to the special relativity for Lorentz then Einstein in 1905, and later to general relativity in 1915.
Even if the results were available, one factor working against Newton was his choice of notation in Calculus. He used the "fluxion" notations, what some called the "dot" notation, where x-dot was the derivative of x with respect to time, x-double-dot was the second derivative. In contrast, Leibniz's notation e.g. dx/dt treats time much more like any other variable. Newton and the English math establishment resisted the adoptation of continental notation into the early 1800s. The use of time as just another dimension was only possible after the abandonment of Newton's notation.