No, domain names themselves do not grant rights. However, owners of older trade names and trademarks can take action against the use of a conflicting domain name. The conflict is then assessed based on trade name law or trademark law.
The use of a domain name can establish trade name rights. This may occur if a business presents itself to the public under the name of the website. Naturally, it is essential to consider the activities the business offers and whether this creates issues under trade name law.
Action is often taken based on trademark law. This can happen if the domain name includes the same trademark or if counterfeit products are being offered on the website.