In short no.
Of course that depends on what you consider is a web designer. I've always considered them to be the people that mock-up the look and feel of a website. You should be able to crop a few background images in photoshop and that's about as much photoshop as you need to learn. In any reasonable workplace there will be a designer who mocks up how the website will finally look. It will most likely be your job to turn this image into code (html/css).
CSS and HTML are important, the people who are creative enough to design pretty pictures in photoshop are usually not good at CSS. CSS is a little like XPATH, you select xml nodes and then apply rules to them.
You should also know a bit of HTTP, cache headers, difference between POST/GET, content-type headers etc.
There will be some occasions where you will need to use common sense when creating some parts of the user interface so they are accessible and usable.
- That ajax part of the page should probably have a loading icon, even though the designer/client has not mentioned it.
- That search filter should use http GET so the results can be bookmarked.
- Those thumbnails should not be resized by just using a width/height attribute on the node. You should create thumbnails on the server.
- That email field should probably use type="email" so mobile devices can display the appropriate keyboard variant.
As an aside, PHP is a lousy language, try and find a non-PHP job if you can.