Christianity was doing just fine prior to any influence of the Roman empire.
Tammuz was the meshing of paganism the RCC did, fact. Actually Tammuz was of Babylonian origin (hint not Arab). Babylon was in modern day IRAQ and dates to Sumerian times.
Tammuz (deity) - Wikipedia
How December 25 Became Christmas - Biblical Archaeology Society
The extrabiblical evidence from the first and second century is equally spare: There is no mention of birth celebrations in the writings of early Christian writers such as Irenaeus (c. 130–200) or Tertullian (c. 160–225). Origen of Alexandria (c. 165–264) goes so far as to mock Roman celebrations of birth anniversaries, dismissing them as “pagan” practices—a strong indication that Jesus’ birth was not marked with similar festivities at that place and time.
1 As far as we can tell, Christmas was not celebrated at all at this point.
The earliest mention of December 25 as Jesus’ birthday comes from a mid-fourth-century Roman almanac that lists the death dates of various Christian bishops and martyrs. The first date listed, December 25, is marked:
natus Christus in Betleem Judeae: “Christ was born in Bethlehem of Judea.”
3 In about 400 C.E., Augustine of Hippo mentions a local dissident Christian group, the Donatists, who apparently kept Christmas festivals on December 25, but refused to celebrate the Epiphany on January 6, regarding it as an innovation. Since the Donatist group only emerged during the persecution under Diocletian in 312 C.E. and then remained stubbornly attached to the practices of that moment in time, they seem to represent an older North African Christian tradition.