Calculating Ka values from std. state thermodynamic data
The definition of $K_a$ for $\ce{HNO3}$ is $$\frac{[\ce{H3O+}][\ce{NO3-}]}{[\ce{HNO3}]}$$ (concentrations in mol/L), which is precisely the $K$ that you get from the $\Delta G_r^0 = -RT\ln K$ formula, assuming you use the right free energies (which is not a trivial matter since they must take into account the interaction of each species with surrounding water molecules, an effect known as solvation).

You may be wondering why the concentration of water does not appear in the equation for $K_a$ (or more generally the equation for $K$ in a reaction in aqueous solution).  Because that concentration remains essentially unchanged throughout, it is kept out of the defining expression for $K$, for simplicity. From the point of view of $G^0$ this is correct since the standard state to which the $G^0$ of water refers is pure liquid water, while the standard state for the other three species is 1M concentration. 


  There's already a 1 molar concentration of hydronium ion IN solution under standard state conditions.


You're correct that for specifying the $G^0$ of $\ce{H3O+}$ the standard state is 1M. But the standard state of $\ce{H3O+}$ is not where the reaction takes place. The same $K$ will be found whether you add the $\ce{HNO3}$ to pure water or to a solution of $\ce{HCl}$ of pH zero -- the final concentrations of $\ce{H+}$, $\ce{NO3-}$ and $\ce{HNO3}$ will be different in the two cases, but $K$ will be the same.



You can find a good discussion of the "standard state" for ionic species in solution here: 
http://www.public.asu.edu/~laserweb/woodbury/classes/chm341/lecture_set10/lecture10.doc

I'll quote a couple of paragraphs.  You can see from this discussion that the calculation of these standard free energies is based exactly on the equilibrium constants, so the formula your professor used works almost by definition!


  As always, we can talk about the Gibbs free energy, enthalpy and entropy of such reactions. However, if we try to break these reactions down into the Gibbs energy, enthalpy and entropy of formation of each species in the reaction, we end up defining things like the Gibbs energy of a sodium cation in solution. Yet there is no way to generate just a sodium cation in solution (that would violate charge balance). Thus, we can only determine the Gibbs energy of formation of pairs of ions. This does not cause any real physical chemistry problems, but it causes a book keeping problem: how do we write down the Gibbs energy of formation of a Sodium cation in a book so that we can use it in any reaction where sodium cations are formed?
  
  The answer is that we come up with a sort of standard ion, $\ce{H+}$ (really something more like $\ce{ H3O+}$) and we define the free energy of formation of all other ions relative to this one. For example, I can measure the Gibbs energy change when I dissolve HCl in water to form $\ce{H+}$ and $\ce{Cl-}$.
  $$\ce{HCl}\text{(g)}\to\ce{H+}\text{(aq)}+\ce{Cl-}\text{(aq)}
 \quad\Delta G^0{}_1
$$
  I can now just define the Gibbs energy of formation of $\ce{H+}$ as zero and then determine the relative Gibbs energy of formation of $\ce{Cl-}$.
  $$\begin{align}&amp;
\Delta G^0{}_1=\Delta G^0_{\mathrm{F}}(\ce{H+})+\Delta G^0_{\mathrm{F}}(\ce{Cl-})-\Delta G^0_{\mathrm{F}}(\ce{HCl})
\\&amp;
\Delta G^0_{\mathrm{F}}(\ce{H+})=0\quad\text{(definition)}\\&amp;
\Delta G^0_{\mathrm{F}}(\ce{Cl-})= \Delta G^0{}_1+\Delta G^0_{\mathrm{F}}(\ce{HCl})
\end{align}
$$