What is the historical explanation behind the name of the French village Ramatuelle?

No, there is no accepted historical explanation for the name at all — and therefore one should exercise caution to firmly assert that the name "is of Arabic origin". It may be, but it seems that other theories over this uncertainty issue have more support according to French Wikipedia.

It is true that the name is quite h*m*phonic to the Arab 'mercy of God' and there was indeed an Umayyad/Saracen presence during the 9th century.

But this folk etymology is a mere isolated hypothesis without much proof except for the Arabic speaking presence in that area and that very h*m*phony. The Region Provence-Alpes-Côte d’Azur, on a map almost 'close to Italy', wasn't anywhere like 'long' under Saracen control, but rather short-lived, like 80 years or so for the region and 60 years for the very place.

The theory was primarily promoted rather late by an Algerian-French Arabist named Évariste Lévi-Provençal.

As such — a legend — it is repeated on the tourist net presence of the locality:

Certains historiens pensent donc que le nom de Ramatuelle pourrait être issu de l’expression arabe "Ramat'Ullah" qui signifie "Providence divine" ou "Rahmat Allah", "la grâce de Dieu".
__[Some historians therefore believe that the name Ramatuelle may come from the Arabic expression "Ramat'Ullah" which means "Divine Providence" or "Rahmat Allah", "the grace of God".]

The place is first attested from 1056 as already Ramatuella.
Compared with رحمه الله /ra.ħi.ma.hu‿ɫ.ɫaː.hu/ we do not only see/hear similarities but also differences.

From the 18th century onwards attempts were made to connect this name to Camatulliques from Camatullici near Toulon — with obvious problems for phonetic shifts from 'c' to 'r'. While certainly not impossible, that isn't very likely either.

Concurrently, hypotheses were put forward that tried a Latin or Greek toponymic derivation from rama (Latin: oar, Greek hard/forced labour).
However, Latin 'oar' would be remus and the -tuella element would remain unconnected and unexplained in both those theories.

A seemingly heavy blow to the Arab-hypothesis would be that the other places known to have been under Arab control for a time were named for example Jabal al-Qilâl (mountains of peaks) und Farakhshinit. But for the first case that name didn't stuck, and in the second it can be traced back to Latin roots, like Farakhshinit to Fraxinetu (modern Freinet) as Latin (from fraxinus) for 'ash grove'.

A more probable hypothesis would be that Ramatu- is of 'some sort' of Gallo-Italic Ligurian substrate origin: too obscure and badly documented to be analyzed and explained completely, but otherwise a good match for some other toponyms in the area.

— Based on French WP: Ramatuelle#Toponymie (The Arab Wikipedia version leaves zero doubt on the theophoric origin…)

For the toponomy of Saracen places in that region compared to contemporary and later French names, as well as the uncertainty involved overall for lack of hard source evidence:

Sénac’s work epitomizes the current historiographical trend, presenting the most relentless argument for a political, military, and/or economic connection between these two cultures [al-Andalus/Saracen Provence, LLC].
His repeated emphasis on the evidence of Arabic geographers like al-Istakhrî (Kitâb al-Masâlik wa al-Mamâlik), Ibn Hawqal (Kitâb sûrat al-ard), yâqût (Mu‘jâm al-Buldân), and Ibn Hayyân (Muqtabas), for example, attempts to reveal the full geographic and human dimensions to Muslim settlement on Provence (referred to in these sources as Jabal al-Qilâl and Farakhshinît).
[…]
Notwithstanding these different disciplinary lenses, the ‘hard evidence’ for tenth-century Provence is still recognizably scarce. Even the exact location of Muslim settlement remains uncertain, leaving only a vague idea of the political centre from which the Saracens exercised their rule for over eight decades.
According to contemporary accounts, Fraxinetum was either: (a) a ‘place [locus] on the shore of the sea, in the province of Arles’; (b) a ‘village [villa] which is known to lie on the border of the Italians and the Provencals’;(c) a ‘valley’ (valles); (d) a ‘château’ (castrum); (e) or not a place (bourg) at all but a wider region in the county of Fréjus (in comitatu Forojuliense). Within this wider enclave (Fraxinetum territorium) of Muslim settlement, therefore, was a castrum, villa, or oppidum near La Garde-Freinet, in the territory of Saint-Tropez, in what is today known as the Massif des Maures.
[…]
As Sénac has argued, we should imagine numerous centre[s] d’habitat72 — a Saracen stronghold comprising the communes of Cucullinus (Cogolin), Sancti Poncii (Saint-Pons les mûres), Roca de Miramars (Notre-Dame de Miramar), Ramatuella (Ramatuelle), Sancti Torpetis (Saint-Tropez), Pampalonam (anse de Pamepelonne), Castri Boriani (le Bourrian), Grimald (Grimaud), and Questa (Notre-Dame de la Queste).

— Kriston R. Rennie: "The Saracen Legend of Tenth-Century Provence", The Mediaeval Journal, Volume 6, Issue 2, 2016 (pp1–24, doi)

Note that Sénac's work cited above is indeed one of the rather few examples supportive for the "probable" Arab origin of the current place name.

A local archaeological society however explains it as a village that

is said to have taken its name from a Celto-Ligurian tribe, the "Camatulici", who inhabited the coastline and its heights and extended as far as the mouth of the Argens.

According to other sources, Ramatuelle, which was occupied by the Saracens, is a deformation of the Arabic Rahmatu'llah ("blessing of God").

However, Pliny the Elder, who cited the village under the name of "Regis Camatullicorum", placed the Camatulici on the mountains to the north of the district of Toulon and thought that this people, allied to the Bormani (coastal people), occupied the coastal heights in times of war.

The Romans succeeded the Celts. They settled in the Salettes district. The old farm of Barraques is partly of Roman construction and brick tombs and medals were found there, one of which bears the effigy of Faustina, Antonin's wife.

At the end of the 9th century, the village was occupied by the Saracens for almost 60 years, an occupation that left few archaeological traces. They built a fort at the top of the Cugullières mountain between Ramatuelle and Gassin. They were expelled in the 10th century by the first Count of Provence, Guillaume I.

— Commune de Ramatuelle P.L.U.: "Rapport de Presentation Pièce n°1", mai 2006. (PDF)

Which leaves us with as little, very little, evidence on or in the ground as in written sources. But we notice that the still standing "Saracen gateway" is of later than Saracen control Medieval construction, and from clearly 'European' builders. Much later then, playing with that Saracen legend seems to be indeed transformed into a tradition:

The main historic sight in Ramatuelle is the Church of Notre Dame which was built in the 16th century against the protective ramparts. Its bell-tower doubled up as a look-out post. Also in Ramatuelle you can see a rather intriguing prison: it is tiny and built in a distinctly Arabian style. It was built by Napoleon III but many believed it was really a Turkish bath built by the Saracens.

— France this Way: Ramatuelle travel guide