Kepler's third law states $C=\frac{r^3}{T^2}$, as seen from the orbits of planets around the sun. But as the earth also attracts the sun, the sun must also orbit the earth and so $c=\frac{r^3}{T'^2}$ from which we can derive $T'^2=\frac{M}{m}T^2$ or $T'=\sqrt{333000}T$ or about 577 year. If this reasoning is correct how can we observe this rotation. But if it is not correct how can we then apply the third law of Newton to derive the law of gravitation?


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You are right. Kepler's third law, as quoted in your question is not exactly true.

The situation in the two-body problem (sun and one planet) is more complicated, because both the sun and the planet orbit around their common barycenter (the $\color{red}{+}$ in the animation below).

(animated image from Wikipedia: Barycenter - Gallery)

The correct form of Kepler's third law for the two-body problem (sun + one planet) can be derived from Newton's gravitational law. And the result is:

$$\frac{a^3}{T^2}=\frac{G(M+m)}{4\pi^2}$$ where
$a$ is the semi-major axis of the elliptical relative motion of one mass relative to the other,
$T$ is the period of the orbit,
$M$ is the mass of the sun,
$m$ is the mass of the planet,
$G$ is Newton's gravitational constant.

Then, because the mass of the sun is so much larger than the mass of the planet (i.e. $M\gg m$), we can use the approximation $$\frac{a^3}{T^2}\approx\frac{GM}{4\pi^2}.$$


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