Did the Catholic Church have a police force in the medieval period?

No, the Catholic Church did not have a Police force around 500 AD to arrest a British King.

Some of the reasons why this was not so:

• Police, as it is known today, started to form during the 17th Century
  • in the Roman Empire the army was commonly used where needed
• The Catholic Church, as a state authority, did not yet exist
  • Kingdoms had envolved from the Western Roman Empire
    • not all of which were Christian (or in name only)
• British Kings (not even English Kings) existed at that time

See Wikipedia quotes below for more details.

If you are interested in how the Catholic Church (Popes) tried to influence Kingdoms outside of Italy, the the best starting point would be research on

• Henry VIII of England
• Francis I of France
• Charles V, Holy Roman Emperor
• Inquisition - Early Modern European history

between 1515 and 1547.

Anglo-Saxon England was early medieval England, existing from the 5th to the 11th centuries from the end of Roman Britain until the Norman conquest in 1066. It consisted of various Anglo-Saxon kingdoms until 927 when it was united as the Kingdom of England by King Æthelstan (r. 927–939). It became part of the short-lived North Sea Empire of Cnut the Great, a personal union between England, Denmark and Norway in the 11th century.

List of British monarchs
There have been 12 monarchs of the Kingdom of Great Britain and the United Kingdom (see Monarchy of the United Kingdom) since the merger of the Kingdom of England and the Kingdom of Scotland on 1 May 1707. England and Scotland had been in personal union under the House of Stuart since 24 March 1603.

State church of the Roman Empire Late antiquity
By the time Christianity became the state religion of the Empire at the end of the 4th century, scholars in the West had largely abandoned Greek in favor of Latin. Even the Church in Rome, where Greek continued to be used in the liturgy longer than in the provinces, abandoned Greek. Jerome's Vulgate had begun to replace the older Latin translations of the Bible.

The 5th century would see further fracturing of the Church. Emperor Theodosius II called two synods in Ephesus, one in 431 and one in 449, the first of which condemned the teachings of Patriarch Nestorius of Constantinople, while the second supported the teachings of Eutyches against Archbishop Flavian of Constantinople.

Police - History - Ancient policing
In the Roman empire, the army, rather than a dedicated police organization, provided security. Local watchmen were hired by cities to provide some extra security. Magistrates such as procurators fiscal and quaestors investigated crimes. There was no concept of public prosecution, so victims of crime or their families had to organize and manage the prosecution themselves.

Under the reign of Augustus [60 AD] , when the capital had grown to almost one million inhabitants, 14 wards were created; the wards were protected by seven squads of 1,000 men called "vigiles", who acted as firemen and nightwatchmen. Their duties included apprehending thieves and robbers and capturing runaway slaves. The vigiles were supported by the Urban Cohorts who acted as a heavy-duty anti-riot force and even the Praetorian Guard if necessary.
Post-classical policing

• first year meantioned here is 1479

Fall of the Western Roman Empire - From 476; last Emperor, rump states
By convention, the Western Roman Empire is deemed to have ended on 4 September 476, when Odoacer deposed Romulus Augustulus and proclaimed himself ruler of Italy, but this convention is subject to many qualifications. In Roman constitutional theory, the Empire was still simply united under one emperor, implying no abandonment of territorial claims. In areas where the convulsions of the dying Empire had made organized self-defence legitimate, rump states continued under some form of Roman rule after 476. Julius Nepos still claimed to be Emperor of the West and controlled Dalmatia until his murder in 480. Syagrius son of Aegidius ruled the Domain of Soissons until his murder in 487. The indigenous inhabitants of Mauretania developed kingdoms of their own, independent of the Vandals, with strong Roman traits. They again sought Imperial recognition with the reconquests of Justinian I, and they put up effective resistance to the Muslim conquest of the Maghreb. While the civitates of Britannia sank into a level of material development inferior even to their pre-Roman Iron Age ancestors, they maintained identifiably Roman traits for some time, and they continued to look to their own defence as Honorius had authorized.

Pope - Middle Ages
After the fall of the Western Roman Empire, the pope served as a source of authority and continuity. Pope Gregory I (c 540–604) administered the church with strict reform. From an ancient senatorial family, Gregory worked with the stern judgement and discipline typical of ancient Roman rule. Theologically, he represents the shift from the classical to the medieval outlook; his popular writings are full of dramatic miracles, potent relics, demons, angels, ghosts, and the approaching end of the world.

Sources:

• History of Anglo-Saxon England
• List of British monarchs
• State church of the Roman Empire
• Catholic Church
• Police - History - Ancient policing
• Pope - Middle Ages
• Fall of the Western Roman Empire - From 476; last Emperor, rump states
• Henry VIII of England
• Francis I of France
• Charles V, Holy Roman Emperor
• Inquisition - Early Modern European history

Question: Did the Catholic Church have a police force in the medieval period? For example, if a British king was accused of heresy by the church, who would have arrested him? I’m specifically interested in the time period around 500 AD

Britain wasn't primarily a Christian territory in 500 AD. Rome had recalled it's legions 383 AD and the Britons had expelled the last Roman magistrates in 404; leaving warlords in charge and Christianity taking a back seat to Paganism in Britain. Arguable Augustine's mission in 597 AD from the Pope to King Aethelbert of Kent is what really set up the future course of Christianity in Britain.

Beyond Britain, the Catholic Church wasn't a temporal power in the time period you specified, 500 AD. In September 476 AD, the last Roman emperor of the west, Romulus Augustulus, was deposed by Odovacar (a Germanic prince); So the primary concern of the Catholic Church in the time period you specified was to promote Christianity and thus their own relevancy among the non Roman barbarians. In this time period the Catholic Church only had power through patronage of the state, (Rome, or Barbarian Kings) and not over the state.

The Catholic Church really didn't become a temporal power util after the Donation of Pepin 756 AD and the coronation of Charlemagne 800 AD.

The Donation of Pepen, granted the Church / Pope rule over the Papal States which became the power base of the Catholic church and allowed the rule of the Popes to extend beyond the duchy of Rome. It made the Pope himself a temporal prince, not just a spiritual leader.

The coronation of Charlemagne of the Franks as the King of All Christendom was the second important trans-formative event for the Catholic Church. Transforming the church from a spiritual institution into a temporal power, and eventually super power. The Pope placed the crown on Charlemagne's head, Making Charlemagne the most powerful king in Christendom but also demonstrating for all that the Pope and Church symbolically were over the King. Over all Christian Kings of Europe. This eventually lead to Monarch's gaining their authority to rule from the Pope/Church God's representative on earth.

This lead to the Pope and Vatican becoming a superpower for a time, conducting Holy Wars and directing Monarchs. The Pope really had no need for a "police force", he had control over armies. If a monarch broke with Rome, the Pope's powerful weapon over the Monarch was to excommunicate him and thus remove his God given authority to rule. What the Pope gives the Pope can take away. Only the strongest European rulers were in a position to resist Rome for much of the middle ages, prior to the Reformation.

From @DenisNardin: While this is not completely false, it should be mentioned that during the Lombard domination of Italy (so roughly from 568 to 756), the Pope was the unofficial head of the duchy of Rome, and so had an army at his disposal, although it was always under the supervision of the Roman emperors (not strictly so, due to the distance and the other problems of the Empire). – Denis Nardin 4 h

To say the Pope was the unofficial head of the duchy of Rome and had an army at his disposal, as soft as that statement is oversteps the Pope's authority. The pope was a respected, influential, even feared spiritual leader. The Duche of Rome was a Byzantium construct and had Greek/Byzantine soldiers, generals, and ministers answerable to the Byzantium Emperor who ruled there.

Even So the Duche was weak, the lumbards were strong but not unified. The Duche survived not through force of arms but through diplomacy, changing alliances and sometimes threats of spiritual/supernatural retaliation by Christian Saints as conveyed by the Pope.

After the Franks under King Pepin the short, on the Pope's request, defeated the Lumbards in 756, king Pepin gifted the Pope rule in the Dutche of Rome and all the territory once controlled by the defeated Lumbards. This was the beginning of the Papal States and was the beginning of the Pope as a temporal prince with his own lands, army, and subjects. Prior to this the Pope did not rule, he influenced. After this he was a feudal leader of a nation.