Picture of plaque: https://pbs.twimg.com/media/Ch71yJhWEAEjS8r.jpg
When I googled the phrase, it appears to be in rural Oxfordshire, but I couldn't find anything about the onslaught of goths and utilitarians which Rev. White vanquished.
OP, if you don't get a solid answer from anyone here, instead of speculation, you may want to contact the Oxfordshire History Centre. They would probably be the best equipped to explain whether the plaque refers to a specific situation, or just the general cultural milieu of the time.
Without more context, here's a guess:
English architecture (and really, "arts" in general) at the end 18th century underwent an attitudinal shift from the then-prevailing neoclassical to more Romantic styles that drew from medieval and, particularly in the later decades of the 19th century, "national" styles. In England, the early stages of this decades-long change was marked by the rise of Gothic Revival architecture, which borrowed heavily from medieval ecclesiastical structures. While Gothic Revivalism went through many phases and found broad applicability, it was particularly prevalent in church construction.
It was not, however, without it's discontents. Gothic ornamentation is elaborate and some considered too decadent for a religion where humility, order, and conservatism were important touchstones. This plaque could very well refer to an attempt to build over (or in some way eliminate) the church green in question; the "Goths" probably refers to proponents of "Gothic architecture", which, by 1853, was surging in popularity through England. The "utilitarian" phrase probably doesn't refer to "utilitarianism" and is too early to reference the later architectural notion of "utilitarian" structure, but could reference a struggle on the part of Rev. White to preserve the small green from people who wanted to use it for another unknown purpose.
So, in brief: it could be a reference to a struggle on the part of Rev. White to keep somebody from building on the green with "Goths" serving as a reference to an architectural trend, not the medieval peoples, and "utilitarian" possibly a response to an argument by some that the green would be put to better use if developed in some way.
The book Best Tea Shop Walks in Oxfordshire By Julie Meech has a location for it, and it's just been added to Read the Plaque at this page: http://readtheplaque.com/plaque/from-goths-and-utilitarians
