He called her on the pretext of needing more information. He called her on the excuse of needing more information. He called her on the reason of needing more information. The first (He called her on the pretext of needing more information) is the only correct sentence structure of these three. It means he made up a reason to justify calling her, but the reason wasn’t real. (This is what ‘pretext’ means). If you used the noun ‘excuse’ you would have to change the sentence structure. Perhaps, “His excuse for calling her was the need for more information.” (An ‘excuse’ is an explanation designed to justify an action – often one you think you will be judged negatively for. For example, ‘Her excuse for being late was that there was bad traffic.’) If you wanted to use the noun ‘reason’, you’d have to change the sentence structure, too. (Perhaps, “The reason he called her was to get more information.”) A reason is an explanation of why you are doing something.

In your first group, "He called her on the pretext of needing more information." is natural. The others need rewording. For instance the 2nd works with "He called her with the excuse of needing more information." In your 2nd group, both first and second sentences are natural. The 3rd is also natural, but you need to change "a excuse" to "an excuse".