The former Ludlow MP, a fierce critic of the scheme since its inception 13 years ago, is calling for the Prime Minister to make repealing the law one of the top priorities in Britain's Brexit plans.
The warrant gives any EU country the right to demand the arrest of any EU citizen who is wanted by their police. But critics say there is great variation in standards of justice among different EU nations, and it has led to many people being incarcerated abroad on flimsy or non-existent evidence.
Mr Gill, a veteran eurosceptic whose opposition to the Maastricht Treaty rocked John Major's government in the 1990s, says the warrant flies in the face of the more than 800 years of English Law, going back, which says that people should not be detained without evidence.
He cites the case of Jason McGoldrick and Michael Turner, two British businessmen who were held in a notorious former KGB prison in Hungary for four months in relation to a failed business venture.
Mr McGoldrick, who had returned to Britain following the collapse of his business, had just returned from holiday with his pregnant wife when he was arrested at Heathrow Airport in 2009. He was handcuffed and kept in custody, before being swiftly extradited to Hungary to face trial.
A supporter described his appearance before a British court as a sham.
"No British court has the right or power to protect British subjects against the EU’s European Arrest Warrant," said the supporter.
"They are simply processed and handed over at the airport to the foreign jurisdiction which has demanded their arrest, often on unsound evidence or for what would be considered in the UK a civil matter, not a crime.
"On arrival in Hungary, Jason and Mike were handcuffed, led through the streets on leads and thrown into prison.
"They languished there without hope for six months, regularly assaulted by inmates who stole their possessions and living on raw, liquid pork fat and frozen vegetables."
They were eventually freed following the intervention of Euro MP the Earl of Dartmouth, and handed suspended prison sentences. Lord Dartmouth said Hungarian jurisdiction meant they could have potentially been held without charge for up to four years over something which would be dealt with by the civil small claims court in this country.
Tony Blair's government signed up to the European Arrest Warrant scheme with a view to speeding up the extradition process with regards to serious and violent crimes.
At the time, the measures were opposed by David Cameron, who was then a newly elected backbencher.
Writing in his local paper back in December 2001, he described the proposed new laws as “monstrous” and warned that British people would be “no longer safe” from European prosecutors who could lock them up without the slightest shred of evidence.
Referring to the case of 12 British plane spotters who were locked up by Greek authorities on suspicion of spying in November 2001, he accused Tony Blair of attempting to “give up rights we have enjoyed for centuries”.
“Our current extradition procedures are not perfect, but they do include safeguards. Before someone is sent to face a court overseas it must be shown there is a case to answer and the Home Secretary must agree. Under the new arrest warrant, these safeguards go.”
Mr Cameron also said, during a parliamentary debate the following year, that UK citizens could be extradited over alleged offences that were not even crimes in this country.
The European Union points out that since the warrant came into force in 2004, it has led to the successful extradition of a failed London bomber caught in Italy, a German serial killer tracked down in Spain, a suspected drug smuggler from Malta who was found in the UK, and a gang of armed robbers wanted in Italy who were arrested in six different EU countries.
But Mr Gill, who lives in Bridgnorth and ran a meat business in Wolverhampton, is particularly concerned about the ease with which other countries are able to secure arrest warrants, based on evidence that would not be acceptable in Britain.
"The system is iniquitous because it authorises arrest without evidence, which is entirely alien to the British tradition of justice with its roots in the law of habeas corpus and the Magna Carta," he says.
Unless the European Arrest Warrant system is scrapped or radically amended, even after the UK has left the EU British subjects will still be subject to false accusation, arbitrary arrest and wrongful imprisonment by dint of warrants arrested by the powers-that-be in foreign jurisdictions."
Mr Gill also quotes the case of Colin Dines, a retired judge who was arrested under a European Arrest Warrant in 2010 on allegations of a link to a telecoms fraud in Italy.
Mr Dines, who was 68 at the time of his arrest, had his warrant suspended after he suffered a stroke while awaiting extradition to Italy.
Mr Gill's successor as MP for Ludlow, Philip Dunne, defends the use of the warrants, saying they are an important way of preventing criminals from avoiding charges by fleeing abroad.
"It has enabled us to secure the extradition of individuals wanted by the police," he says.
"We have deported individuals wanted by others, it has led to successful co-operation."
Mr Dunne disputes the claim that it can be used to enable the arrest of British citizens without evidence.
"I do not believe this is how it works in practice since British police forces work with their counterparts from other EU countries in exercising this power."