Gfrktyl

According to Newton's third law of motion that states that every action has an equal and opposite reaction.

So, if the Earth exerts a gravitational pull on us (people) then even we should exert a force equal and opposite (in terms of direction) on the Earth.

It is intuitive to think that this force is really small to get the Earth to move. But, if we take a look at the second law of motion that states that F = ma we see that however small the force, there will be some amount of acceleration. Therefore, even though we exert a very small gravitational force on the Earth it should be enough to get the Earth to move even though the acceleration is a very small amount.

But what I said clearly does not happen. So there must be some flaw with my reasoning. What is that flaw?

The acceleration that your gravitational pull causes in the Earth is tiny, tiny, tiny because the Earth's mass is enormous. If your mass is, say, $70;mathrm{kg}$, then you cause an acceleration of $aapprox 1.1times 10^{-22};mathrm{m/s^2}$.

A tiny, tiny, tiny acceleration does not necessarily mean a tiny, tiny, tiny speed, since, as you mention in comments, the velocity accumulates. True. It doesn't necessarily mean that - but in this case it does. The speed gained after 1 year at this acceleration is only $vapprox 3.6×10^{-15};mathrm{m/s}$. And after a lifetime of 100 years it is still only around $vapprox 3.6×10^{-13};mathrm{m/s}$.

If all 7.6 billion people on the planet suddenly lifted off of Earth and stayed hanging in the air on the same side of the planet for 100 years, the planet would reach no more than $vapprox 2.8times 10^{-3};mathrm{m/s}$; that is around 3 millimeters-per-second in this obscure scenario of 100 years and billions of people's masses.

Now, with all that being said, note that I have to assumed that all those people are not just standing on the ground - they must be levitating above the ground.

Because, while levitating (i.e. during free-fall), they only exert the gravitational force $F_g$:

$$sum F=maquadLeftrightarrowquad F_g=ma$$

and there is a net acceleration. But when standing on the ground, they also exert a downwards pushing force which is their weight $w$:

$$sum F=maquadLeftrightarrowquad F_g-w=ma$$

Now there are two forces on the Earth, pushing in opposite directions. And in fact, that force called weight exactly equals the gravitational force (because those two are the action-reaction pair from Newton's 3rd law). So the above formula gives zero acceleration. The forces cancel out and nothing accelerates any further.

In general, any system can never accelerate purely by it's own internal forces. If we consider the Earth-and-people as one system, then their gravitational forces on each other are internal. Each part of the system may move individually - the Earth can move towards the people and the free-falling people can move towards the Earth. But the system as a whole - defined by the centre-of-mass of the system - will not move anywhere.

So, the Earth can move a tiny, tiny, tiny bit towards you while you move a much larger distance towards the Earth during your free-fall so the combined centre-of-mass is still stationary. But when standing on the ground, nothing can move because that would require you to break through the ground and move inwards into the Earth. If the Earth was moving but you weren't, then the centre of mass would be moving (accelerating) and that is simply impossible. The system would be gaining kinetic energy without any external energy input; creating free energy out of thin air is just not possible. So this never happens.

Short answer, it does. It's just too small for you to notice. So, why isn't it noticeable?

The earth doesn't noticeably move towards you despite your gravitational pull because you're resting on it, the same way you don't move towards the center of the earth if you have stable footing on its surface.

If you're not standing on it (e.g. skydiving, or in orbit) then the earth does start to move towards you (or, if you jump up, move in the opposite direction of your body). But since the earth's acceleration imparted by you is so very small, although it's calculable, it's basically immeasurable. As @Steeven calculated, a 70Kg person would impart an acceleration of $aapprox 1.1times 10^{-22};mathrm{m/s^2}$ which isn't noticeable with our human perception.

Adding to the very small levels of acceleration, one more reason you don't notice earth's movement is that many, many things happen on earth. It's optimistic/unrealistic to model the earth as a perfectly rigid body, but let's do it for the sake of this example: there's so much stuff happening all around earth, each imparting a small acceleration on the whole body (cranes lifting things, mines being excavated with explosives, landslides and avalanches, earthquakes etc.) that even if we were able to measure the tiny acceleration that you exert on earth it would be near impossible to isolate the impact that you have from all the noise created by everything else.

From newtons third law we know one thing.
Every action has an equal and opposite reaction.
This means that the force we act on earth is equal to the force that the earth acts on us.

This means

f = ma ( by person )

The average mass of a person is 70 kg and acceleration due to gravity is nearly 10 m/s^2.

So the force we apply on earth is nearly 700 N.

Now

F = MA ( by earth)

The aproximate mass of earth is 6 × 10^24 kg.
But the force remains 700 N

Now A = F/M

A = 700 / 6* 10^24.

=> ~ 116 * 10^-24 m/s^2.

That acceleration applied by one person is so miniscule that it does not need to be considered.

The acceleration of earth is in negative powers of 24 . So we need more that just billions of people to accelerate the earth upwards.