The correct way of saying this is: My date stood me up. If you mean that your date did not turn up, the sentence has to contain a pronoun, e.g. her date stood her up/his date stood him up/your date stood you up/my date stood me up. If you say: My date stood up, it means that your date literally stood upright from a sitting position. You would not generally use this phrase in the present tense, although you could say: My date will stand me up /might stand me up. Hope this is helpful.

Shirley's points are all correct. To answer your main question, yes, you can use the phrase "he stood me up" to describe a situation where your date said he would call you, but then didn't call. The phrase isn't normally used this way, though, so you should probably find a way to make it clear that you're talking about a "phone date" and not a "real" date. For example, you could say: "My boyfriend was supposed to call me last night, but he stood me up." Or: "My date stood me up. He said he was going to call, but he never did."