He highlights a lack of detail on EEA membership, the UK's likely liabilities upon leaving the European Union, what regulatory bodies and regimes will need to be replaced upon exit, and says there is "no meaningful detail" on how the government will control migration.
Legal commentator Joshua Rozenberg told Sky News: "What the claimants are saying is they want the court to say the way in which the United Kingdom government could pull out of the single market, if that is what it decides to do".
UK's High Court has blocked a Brexit legal challenge that asserts parliament must approve the country's departure from the European Economic Area (EEA).
British Influence, a Pro-EU British think tank, in November launched the legal case which sought a court-ordered clarification on the extent to which Theresa May's government could interpret the UK's June referendum vote.
The application was brought by Peter Wilding, the head of pressure group British Influence, and Tory lobbyist Adrian Yalland.
Refusing to give the go-ahead to the new challenge, the judges said they would giver their reasons later.
Following their conversation, in which she updated Mr Rajoy and Mr Kern on the UK's preparations for Brexit, Mr Rajoy said her speech "clarified many things".
James Eadie, prosecuting for the government, told the hearing that since ministers had not yet decided which legal route would be taken to leave the EEA, then the challenge was "premature" and should be rejected. The government has a mandate, not a blank cheque.
Minister for UK Negotiations on Scotland's Place in Europe Michael Russell said: "The people of Scotland did not vote for Brexit, and only one of the nation's 59 MPs has now backed the UK Government by voting for the triggering Article 50".
"We are glad this attempt. has been dismissed", he said in a statement.
Parliament is in the process of considering legislation which would give Theresa May the authority to notify the European Union of the UK's intention to leave by the end of March.