Yes and No.
Yes you can keep your java business logic and data handling code (presumed you did a clean separation of your application layers). E.g. just convert to a standalone java app and you're fine.
No, if you look for something to run in the webbrowser then you have to migrate to JavaScript/HTML/JSF/GWT... on the frontend and your java code running on the backend (JEE or Servlets ...).
Regarding the "No", it depends on what kind of application you have. A web application or a standalone application?
(1) For a web application you have to recreate the front end part (UI, communication with the server) but you can reuse parts of your logic.
(2) For a standalone application (using Swing/AWT/JavaFX ...) you can keep your code, just create a standalone app out of it. For distribution e.g. JavaFX offers an application packaging mechanism.
PS: The quickest way could be the conversion to the webstart technology however it was also deprecated and will be also be removed in future. So I would not use it.