Is it necessary to re-read all the supplications or Quranic verses that were recited, or anything else venerated in Islamic law, after realizing that the mobile phone was held while impure during internet browsing?

You are not required to reread what you have already recited from the Quran. However, you should recite it in the most complete and best states, such as being keen on the purity of your clothes, body, and mouth. It is disliked to recite the Quran while the mouth is impure before washing it, but the more correct opinion is that it is not forbidden.