Complex Morlet function and DWT
A continuous wavelet frame can be discretized with perfect reconstruction under some technical conditions. If this is possible, you obtain a discrete wavelet frame, often with some amount of  redundancy (depending on the wavelet shape). This is a result formalized by I. Daubechies, for instance. 

For a long time, apart from the Haar or the Franklin wavelets, little was known about the possibility of obtaining a critical (non-redundant) discrete wavelet transform. Until the inception of the Meyer wavelet, and the subsequent ones (Daubechies, Symmlets, Coiflets, etc.). Their somewhat weird shapes is the sign that not every wavelet shape is possible. This is even truer as the complex Morlet is complex, hence somewhat already twice redundant for real signals. 

So, Morlet cannot be critically discretized with 2-band (dyadic), as you already guessed, as you wanted to compare CWT and DWT. However, multi-band  filter banks are an option to build orthogonal counterparts of "Morlet"-like design. Malvar wavelets, Modulated complex lapped transforms or Lapped orthogonal transforms are  instances (see a Panorama of wavelets for references), and their can be  further decomposed across scales as $M$-band wavelets. 

Or you can accept a non perfect reconstruction dyadic decomposition akin to Morlet.