Can I execute an ILS approach using a NAV CDI with LOC signal only?
No, you cannot execute an ILS approach without an ILS glide slope indication as the primary form of vertical guidance.

Even if you could supplement it with alternatives (WAAS, GPS software, etc), they are not recognized as the appropriate standard of vertical guidance required for an ILS approach.

Many airports that have an ILS also have LOC (localizer-only) approaches.  If such an approach is published, you could certainly fly it with only the use of the localizer, but then again, it is not an "ILS approach" it is a localizer approach.

The localizer -only approach will have different minimums which will be based on your altitude at a published minimum descent altitude rather than a decision height.

If no localizer approach is published, then you'd need to find another method or airport.