B

Step 1: We start with the formula for gravitational force between two masses, which is given by Newton's law of gravitation:
$$ F = G \frac{m_1 m_2}{r^2} $$
where:
- $F$ is the gravitational force,
- $G$ is the gravitational constant (approximately $6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2$),
- $m_1$ and $m_2$ are the masses of the two spheres,
- $r$ is the distance between the centers of the two spheres.

Step 2: Since the spheres are of the same material and have the same radius 'r', we can calculate their masses using the formula for the volume of a sphere and the density:
$$ V = \frac{4}{3} \pi r^3 $$
The mass of each sphere can then be written as:
$$ m = V \cdot \rho = \frac{4}{3} \pi r^3 \cdot \rho $$
Thus, the masses of both spheres are:
$$ m_1 = m_2 = m = \frac{4}{3} \pi r^3 \cdot \rho $$

Step 3: The distance between the centers of the spheres is equal to the sum of their radii, which is:
$$ r_{ ext{center}} = r + r = 2r $$

Step 4: Substituting the values of masses and the distance into the gravitational force formula gives:
$$ F = G \frac{\left( \frac{4}{3} \pi r^3 \cdot \rho \right) \left( \frac{4}{3} \pi r^3 \cdot \rho \right)}{(2r)^2} $$
Simplifying this:
$$ F = G \frac{\left( \frac{16}{9} \pi^2 r^6 \rho^2 \right)}{4r^2} = G \frac{4 \pi^2 r^4 \rho^2}{9} $$

Step 5: Thus, the gravitational force between the two spheres is:
$$ F = \frac{4}{9} \pi^2 G r^4 \rho^2 $$
Therefore, the correct answer option is 'B'.