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How did Newton predict the inverse-square rule?

By the 16th century, sufficient data on the movement of planets had been collected by many astronomers. Based on those data, Johannes Kepler derived three laws, which govern the motion of planets. These are called Kepler's laws.
They are

Kepler's first law:

The orbit of a planet is an ellipse with the sun at one of its foci.

Kepler's second law:

The line joining the sun and the planet sweeps out equal areas in equal intervals of time. Therefore, if the travel time from $$A$$ to $$B$$ is the same as from $$C$$ to $$D$$, then $$OAB$$ and $$OCD$$ are equal.

Kepler's third law:

The cube of a planet's mean distance from the sun is directly proportional to the square of its orbital period ($$T$$).

It is mathematically given by,

$\begin{array}{l}{r}^{3}\propto {T}^{2}\\ \frac{{r}^{3}}{{T}^{2}}=\mathit{Constant}\end{array}$

Assume that the planetary orbits are circular. Assume the orbital velocity is $$v$$, and the radius of the orbit is $$r$$.

Then the force acting on an orbiting planet is given by

$F=\frac{m{v}^{2}}{r}\to 1$

If T is time period, then its velocity is given by

$\begin{array}{l}v=\frac{2\mathrm{\pi }r}{T}\\ \mathit{This}\phantom{\rule{0.147em}{0ex}}\mathit{can}\phantom{\rule{0.147em}{0ex}}\mathit{be}\phantom{\rule{0.147em}{0ex}}\mathit{rewritten}\phantom{\rule{0.147em}{0ex}}\mathit{as},\\ {v}^{2}\propto \frac{1}{r}×\frac{{r}^{3}}{{T}^{2}}\\ \mathit{Since},\frac{{r}^{3}}{{T}^{2}}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{constant}\phantom{\rule{0.147em}{0ex}}\mathit{by}\phantom{\rule{0.147em}{0ex}}\mathit{Kepler}’s\phantom{\rule{0.147em}{0ex}}\mathit{third}\phantom{\rule{0.147em}{0ex}}\mathit{law},\\ \mathit{Therefore},\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{equation}\phantom{\rule{0.147em}{0ex}}\mathit{becomes}\\ {v}^{2}\propto \frac{1}{r}\to 2\end{array}$

Combing, the equation $$1$$ and $$2$$

$\begin{array}{l}F\propto \frac{{v}^{2}}{r}\left(\mathit{We}\phantom{\rule{0.147em}{0ex}}\mathit{know},{v}^{2}\propto \frac{1}{r}\right)\\ F\propto \frac{1}{{r}^{2}}\end{array}$

Thus, Newton got the inverse square rule from Kepler.