Let's say you're a business in the EU that needs a sales management system. In it, you store personal data of your clients. You are considered a "controller"; the SaaS sales management system (that will store this data for you) is considered a "processor."
GDPR states that to remain compliant, Controllers must only use GDPR-compliant Processors.[1]
Assuming that EU startups will take GDPR more seriously than non-EU companies (a safe assumption judging from the posts here on HN), non-tech EU companies (Controllers) will tend to gravitate towards EU tech companies (Processors) to ensure they remain compliant.
It is non-EU (primarily US) tech companies' game to lose. If US tech companies get a reputation for not caring about GDPR, there could be a real sea-change in EU buying behavior.
[1] Article 28, paragraph 1: "Where processing is to be carried out on behalf of a controller, the controller shall use only processors providing sufficient guarantees to implement appropriate technical and organisational measures in such a manner that processing will meet the requirements of this Regulation and ensure the protection of the rights of the data subject."