Plot of Ic and Vbe function
There has to be a mistake in your calculations....

At 300 kelvin kT/q = 26 mV so \$e^{\frac{0.6V}{0.026V}}\$ = 1.05E10

Multiply this by 1E-16 and I get a current of 1.05 uA or very nearly the same as the proper graph.