Short answer: no
A trademark only carries legal weight in the jurisdiction that granted it. A European Union trademark, for instance, covers the EU member countries and nothing beyond them. Think of a company incorporated in one particular state: that state's authority stops at its borders, and a country likewise cannot stretch trademark rights past its own territory. Even with the various international treaties and agreements in place, no single worldwide trademark right exists. Our advice is to register your mark in every jurisdiction where you already do business or expect to do so soon.
Where do the Madrid System and WIPO fit in?
Many customers ask about filing internationally through the Madrid System. It lets you take a trademark application from your home country, which must be a signatory, and extend it to other signatory countries. Each extension, though, is handled as its own separate application and follows the rules and procedures of the country involved.
That matters if an opposition or objection comes up. In that case a local attorney has to respond, because the attorney in your home country cannot do so on your behalf.
Applications filed through WIPO can also cost more, so we generally suggest filing country by country. If you intend to apply in a large number of countries at the same time, however, the Madrid System may be the better fit.