GDPR compliance is a matter between every customer and the business, not between different customers.
How did you get the other customer's contact details? If they were provided or leaked by the business, that might be a failure of the business's obligation as a data controller to protect the personal data they are processing, possibly even a data breach in the sense of the GDPR.
When you contacted the other person this was presumably a purely private or household activity, just like contacting any other personal acquaintance. In that case, the GDPR simply does not apply to any “processing” you may have done (compare Art 2(2)(c)). Things would be very different if you were promoting your own business, but that doesn't seem to have been the case.
If the matter really is as plain as you described, then you can effectively ignore their references to the GDPR until you are contacted by your data protection authority, by their lawyer, or are served with court papers. None of these should happen: even if you were a “data controller” and your processing were subject to the GDPR – only the other customer and not the business would be the data subject, and only a data subject has a right for remedies like making complaints to the data protection agency and only the data subject would have standing to sue you in court for GDPR violations.