Relationship between band gap and built in potential for PN Junction Diode in equilibrium?
No --- the built-in potential is equal to the difference of the Fermi levels in bulk semiconductor N and P, so for a given material it basically depends on the doping of the P and N zones. See the figure in In band diagram, why the Fermi energy (EF) is constant along the device?

Think about it --- if you were right, all the diodes of a given material would have the same built-in potential. 

So the answer is that (using the simply common model of the diode): 

$$
V_0 = \frac{k_BT}{q_e} log (\frac{N_AN_D}{n_i^2}) 
$$

with the usual meaning of the symbols. Notice though that in the term \$n_i^2\$ lie a dependency to the band-gap of the material, too. 

(And be sure to not mistake the built-in potential with the threshold voltage, they are completely different beasts).