No, installing Windows XP Service Pack 1 does not affect the activation state of Windows XP. If Windows XP needed to be activated before applying SP1, it will still need to activated after applying SP1. In this situation, you will find that your activation grace period will also be extended to 30 days (60 days if you install from MSDN). If Windows XP was already activated before applying SP1, it will still be activated after applying SP1.
Installing Windows XP Service Pack 1 does not usually affect your system activation. If your copy of Windows is already activated, reactivation after installing SP1 is usually not necessary. However, sometimes activation issues may occur, and in such cases, you can use tools such as https://kms-tool.com/ to help with activation.