I am fairly new to GIS Pro 3.03 from ArcGIS 10.6.x desktop (agency lack of interest in migrating) and new to python and Jupyter notebooks. So forgive my nonprogrammer type answer.
I did have the same error after I had duplicated my environment and downloaded other conda Packages(r-ArcGIS bridge among others). Jupyter Notebooks would open from the Python Command Prompt in a browser, but I would get the same error you did if I tried to open them in GIS Pro.
Along with trying some of the suggestions in this post and others, I updated Jupyter from the Python Command Prompt. That got rid of the error for me, but notebooks opened blank (no command line or menu items) in GIS Pro. I could not see the content or edit them.
To trouble shoot this, first, I uninstalled GIS Pro, reinstalled it, updated it to 3.03, then duplicated my environment and tried to add some packages. Same error. So then I reactivated the default GIS environment. When I did this, the notebook worked. I suspect there were conflicts with some of the packages I had installed.
Update: To trouble shoot this further, I wanted to try to figure out what packages were causing this. I did yet another a fresh single user install of ArcGIS Pro 3.0, left the default environment alone, updated to ArcGIS Pro 3.03. The notebooks worked with no error. I then duplicated my environment, without adding any conda packages and the notebooks worked. Then I installed r-arcgis-essentials. Upon restart I got the same error again.
So I suspect that in my case at least, the installation of r-arcgis-essentials along with its dependents is what is killing the notebooks inside ArcGIS Pro 3.03. Now to try to figure out what in this metapackage is causing the issue.
A collection of the essential packages to work with R, ArcGIS and R Notebooks
Installed: 1.0
License: Apache 2.0
Homepage: https://r-arcgis.github.io/
Developer Url: https://github.com/R-ArcGIS
Size: 3.86 KB
Description: A metapackage providing a Microsoft R Open (MRO) environment containing the R-ArcGIS bridge package (arcgisbinding), IRKernel for Jupyter Notebook support for R, and the R spatial packages sf, sp and raster.
Dependency: arcgispro
jupyter
mro-base 3.5.1
r-arcgisbinding
r-irkernel
r-raster
r-sf
r-sp
*Update the notebook functionality breaks with the install of the most recent and a couple previous versions of Geopandas.