No, the mRNA vaccines only contain part of the spike protein, the PCR and antigen tests look at a broader set of markers. It could confuse an antibody test, but these are generally only used to assay immunity across a population.
In other news, I managed to book a doctor's appointment last Thursday – for tomorrow, and that only earns me a cherished telephone call. Well, is it urgent? they ask when I query a six-day wait, on the basis God only took that long to create. I did have to rein in the sarcastic response let's see if I answer the phone. (I don't think it is and I'm quantifiably not dead, but I'm not sure of the value of asking patients to triage themselves.)