Has there ever existed a Catholic local church feast at Rome in honor of the Domine Quo Vadis?
This is going to be answer by way of deduction, rather than one supported by an actual source. It should perhaps have been put in the comments, but it would be too long.
Some time ago I asked this question: What date to place the patronal feast day of St. Mary’s Church? Geremia‘s answer truly fit the bill.
Your hunch "that it is the Feast of the Assumption" appears correct, at least according to
Rev. W. J. Wiseman, “The Titular Feast,” The Pastor 3, no. 5 (March 1885): 129–36.
In the § "Practical Directions" (p. 131), he writes:
Blessed Virgin is a general title. […]
(a) If the titulus is simply B. M. V., the titular feast is to be celebrated on the feast of the Assumption, where everything remains as now in the Ordo.
Given that fact that churches dedicated to the Virgin Mary with any fixed date for a patronal feast day are to be that of her Assumption.
It seems logical that the feast day for the Church of Domine Quo Vadis would be that of the major feast day of St. Peter on June 29 (his martyrdom): Feast of Saints Peter and Paul.
Seeing that there is also a direct link to this episode and the martyrdom of the Prince of the Apostles, it again makes sense that it patronal feast day was June 29, until the title of the church changed and became Santa Maria in Palmis.
Chiesa del Domine Quo Vadis is a tiny church near the start of the Appian Way. It was built on the site where St. Peter saw a vision of Jesus while St. Peter was fleeing the persecution of Nero in Rome. Peter asked Jesus, "Lord, where are you going?" (latin: Domine quo vadis?). Jesus answered "I am going to Rome to be crucified again." Peter returned to Rome, where he was crucified in about 64 AD by the emperor Nero in the Circus of Nero, which became the site of St. Peter's Basilica. Inside the nave of the Domine Quo Vadis church, you can see a stone with the footprints of Jesus worn into it. - Domine Quo Vadis
The story seems more legendary than fact, but it remains a sacred place for many pilgrims in Rome.
Church of Domine Quo Vadis?*
Chiesa di Santa Maria in Palmis, better known as Chiesa del Domine Quo Vadis is a small Church southeast of Rome. It is located about 800 m from St. Sebastian Gate (Porta San Sebastiano), where theVia Ardeatina branches off the Appian Way.
There has been a sanctuary on the spot since the 9th century, but the current church is from 1637. The current façade was added in the 17th century.
It has been supposed that the sanctuary might have been even more ancient, perhaps a Christian version of some already existing temple: the church is in fact located just in front of the sacred Campus dedicated to Rediculum, the "God of the return" (his name comes from the Latin verb redeo = to come back); this campus hosted a sanctuary for the cult of the god, that received devotion by travellers before their departure, specially by those who were going to face long and dangerous journeys, towards far places like Egypt, Greece or the East (and the travellers who returned, always stopped to thank the god of the happy outcome of the journey).
The position of the sanctuary in Campus Rediculi was not accidental, first of all because the ancient Appian way was the most important among the Roman "consular" roads, secondarily because from this location the traveller could give the last look to the walls of Rome. In the sacred field there was also the tomb of a famous talking crow, buried at the time of emperor Tiberius with a great popular funeral. Besides, the god Rediculum had a terrible reputation: a legend remembers that Hannibal, after the Battle of Cannae, arrived in front of the doors of Rome following this road; here the god appeared to him in a frightful way, in order to suggest retreating with all his army. This legend lets us suppose that Romans held Rediculum in deep consideration. At the moment we still do not know with precision where exactly the sanctuary was; however an error by some authors of 17th century caused many commonly to think still today that the temple of Rediculum was the sepulchre also said to be of Annia Regilla, toward the inner side of the Caffarella park.
The presence of the Apostle Peter in this area, where he is supposed to have lived, should however find a confirmation in an epigraph in the catacombs of Saint Sebastian, that recites "Domus Petri" (house of Peter) and in an epigram by Pope Damasus I (366-384), in honor of Peter and Paul, in which we can read: "You that are looking for the names of Peter and Paul, You must know that the saints have lived here".
The two footprints on a marble slab at the center of the church (copy of a relief conserved in the near basilica of San Sebastiano) would be the miraculous sign left by Jesus: it is actually a draft of an "ex voto" paid for the good outcome of a travel, what would confirm the supposition that some connection might link the two temples.
The real name of the church, very little known, indeed, is Chiesa di Santa Maria in Palmis, where palmis stands for the soles of Jesus.
It has to be noted that the Roman Catholic Church admits the legend, after Pope Innocent III declared the fact was true; his decree is included in the Decretali di Gregorio IX book IV, tit. 17, cap. Per Venerabilem.
There was an inscription above the front door on the façade, saying: "Stop your walking, traveller, and enter this sacred temple in which you will find the footprint of our Lord Jesus Christ when He met with St. Peter who escaped from the prison. An elemosina for the wax and the oil is recommended in order to free some spirits from Purgatory". Pope Gregory XVI found it so inappropriate (effectively being a sort of advertising) that he ordered its removal in 1845.
In 1983 Pope John Paul II defined the church "a place that has a special importance in the history of Rome and in the history of the Church".
About the foot prints of Our Lord and the in Palmis of the title change:
The church of the “Domine Quo Vadis” is one of the first churches located on the Via Appia Antica, about 800 meters beyond Porta San Sebastiano. The Church has medieval origins, but was rebuilt in 1600. It takes its name from the oral tradition according to which the apostle Peter, fleeing from the city to avoid martyrdom, meets Jesus to whom he addresses the following words “Domine quo vadis (Lord, where are you going)?” And the Lord replied “Venio Romam iterum crucifigi (I am coming to Rome to be crucified again)”. Peter, aware of the rebuke, turns back to face his destiny and Jesus disappears but, in disappearing, he leaves the impressions of his footprints on the road.
As evidence of the incident, within the Church there is a stone with the imprints “of His holy feet”, left by Jesus precisely on the site where the Church now stands. The stone is actually a copy: the original is in fact preserved in the Basilica of San Sebastiano. The second name with which the church is known: Santa Maria “in Palmis” derives from this episode. - “DOMINE QUO VADIS” - SANTA MARIA “IN PALMIS”
In the Early Church only Martyrs were honoured as saints. For this reason the original title of All Saints Day was Sancta Maria ad Martyres. This is in a sense reflected in the title of Sancta Maria ad Palmis.