In a letter of application on Friday, Erdogan asked the Court to ensure that recent court rulings which order the removal of some web content are implemented.
The rulings had demanded that websites - popular micro-blogging site Twitter among others - take down content that causes human rights violations, including against Erdogan and his family.
The Turkish government blocked Twitter last month for two weeks, which was followed by an ongoing ban on video-sharing site YouTube.
In the letter, Erdogan also claimed 50,000 Turkish liras in damages due to a “loss of trust” in judicial rulings.
Earlier, the Constitutional Court rescinded part of a law on Turkey’s top judicial body HYSK that would give more powers to the justice minister. The decision promptedErdogan to accuse the Court of meddling in politics.
- ‘Parallel state’
Wiretapping scandals have made headlines in Turkey since Erdogan announced in late 2012 that a "bug" had been found in his office.
The leaks intensified on the eve of March 30 local elections, following a graft probe that targeted Erdogan’s allies in December last year.
The government decried the wiretappings as a "dirty plot" set in motion by a "parallel state" -- a shadow structure allegedly nestled within the country's police department and judiciary which Erdogan has said is led by U.S.-based Turkish preacher Fethullah Gulen.
Turkey's telecoms authority TIB introduced an "administrative" block on YouTube late March, after a leaked recording of a security meeting of high-ranking officials appeared on the site.
An Ankara court ordered two weeks ago for the lifting of the block after evaluating an appeal by the website's lawyer, Gonenc Gurkaynak.
However, the TIB said on Thursday that it would keep blocking access to YouTube because the company had not complied with Turkish court decisions ordering the removal of certain content.
The Turkish government has been on unfriendly terms with Twitter on similar grounds.
On Monday, a team of senior officials from Twitter's management, led by the company's head of global public policy, Colin Crowell, paid a visit to the Turkish capital of Ankara for a meeting with representatives from the TIB.
It was to discuss current problems between the two parties and the possibility of Twitter opening an office in the country.
Twitter officials said they would act in a faster and more sensitive manner in implementing court decisions,Erdogan's office said. "As a sign of goodwill, they implemented five court decisions ahead of the visit," it added.
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