For a proper ionic bond, you would need ions. However, the backbone of your peptide chain does not have ionic groups. While the resonance structure you drew is correct, note that it is only a resonance structure. It does a good job at explaining the lesser reactivity of an amide bond with respect to an ester or an acid chloride. And it also shows the partial negative charge of the oxygen. But nitrogen is not positively charged in an amide bond (it is slightly negatively charged). Thus, there is no positive counterion that your negative carbonyl oxygen could bond to.
A hydrogen bond is a very viable possibility, though. You may know that it requires three things:
a hydrogen atom (check)
bonded to an electronegative atom (typically $\ce{O, N, F}$ — check)
and another electronegative atom that can receive (typically the same atoms — check)
So a hydrogen bond is possible where an ionic interaction is not.
Note that ionic interactions do happen in peptides. The side chains aspartate and glutamate are deprotonated (negatively charged) at ambient $\mathrm{pH}$ and the residues of lysine and arginine are positively charged. These can now form actual ionic interactions — although typically there will also be a significant hydrogen bonding component in these tertiary structure-creating bonds, too.