Can two planets in an empty universe meet/be pulled together? For this question assume that the entire universe is completely empty. The universe is not expanding or contracting, it is completely motionless since time immemorial and has never moved before.

Only two identical earths without moon are left in this entire universe and they are 5 billion light years apart. From their starting position the earths do not spin or move at all they are completely motionless.

Would these planets ever meet or at least be pulled together?  Or are the their respective gravities too small to have any meaningful effect?
Yes, they would experience gravitational attraction. It would take a long time for them to collide... the formula (derived here and shown here) is:

$$t = \frac{\pi}{2} \sqrt{\frac{d^3}{2G(m_1+m_2)}}$$

where $d$ is the initial distance between the two planets of mass $m_1$ and $m_2$ and $G$ is the Gravitational constant. This gives a time of about $10^{23}$ years, much much longer than the universe has existed. This assumes Newtonian mechanics. Relativity would not change the conclusion much.

There is no known upper limit to gravity, and plenty of indirect evidence that it has no upper limit.