No you do not need vector form for images to look good on other devices. However you need a bigger original image or it will be either upscaled or tiny in a device that has very high pixel density.
See browsers in high density machine have a problem. If they show images at 1 pixel to 1 pixel ratio then all images for legacy websites would be hideously small. So what they do is they override your image size and interpolate more pixels. Now most of these browsers will show the image at exactly the size you specify if you override the image to use in css or with a meta attribute in the header to let the browser know what to do. This may not help as the image will be very small on these devices compared t devices that dont have so dense pixel density.
In reality most likely you just want a separate icon for these devices with more pixels for those screens. And really the only way to do that is t redraw the icon.
But your use case s a bit opaque in the question, its hard to say. anything definitive.
NOTE: PPI/DPI is just a red herring in the story. All that matters is that you have enough pixels. And explain the browser that you are a modern site.