The ultimate font of all these words, including both English rodent and Portuguese roer, is the classical Latin (and also modern Italian) verb rodere/rodo, which in English means “to gnaw or corrode”.
To elaborate somewhat upon the most simple and direct answer, let’s look at what a couple thousand years of hungry rodents gnawing away at our languages have produced.
Of Mice and Muribus
Although compared with the incorrodible classical Latin form, the Italian version hasn’t been gnawed upon at all due to the temerity of Italian mice, in French we find that it’s slightly corroded to just roder instead.
However, down in Iberia where the mice are fiercer, and especially in Portugal whose mice are the fiercest of all Romance mice, it’s eroded even further down to just roer in both Spanish and Portuguese.
But that’s not all. The Iberian mice have chewed off thoroughly irregular portions of nearly all the inflections of this verb there:
Spanish has the infinitive roer go to a regular roo or to either of the equally irregular roigo or royo in the present indicative (nobody can make up their minds which of the three to use) and hence has irregular present subjunctives all the way down if you’re in the 2nd or 3rd camp; preterite 1sg roí but irregular 3sg royó and hence irregular royese etc. in the imperfect subjunctive; irregular past participle roído and irregular gerund royendo.
Portuguese has infinitive roer become in the present 1sg roo (or irregular rôo in Brazil) but irregular 2sg róis, 3sg rói; irregular preterite 1sg roí; irregular past participle roído.
As you see, many of those inflections have been quite irregularly “gnawed upon” by the indigenous Iberian “gnawers” — that is, by their “rodents”.
Meanwhile in the east of Iberia over in Catalunya, things somehow took a slightly different turn when Latin rodere/rodo mutated into the somewhat more exotic form, a Vulgar Latin rosicare derived instead from rosus, the present active participle of rodere/rodo.
And so the Catalans, having milder mice than most of Iberia but nonetheless subjected to the erosive forces of history, produced from that putative Vulgar Latin form the almost surprisingly regular rosegar as the verb and rosegars for the gnawsome critters themselves — or the slightly more chewed-upon rosegaires for their northerly Occitan/Provençal cousins.
Lest it were less than obvious from my agnawing diversion above, the English words that share the same common Latin-derived element include not only the obvious rodent as one who gnaws, but also things like corrode and erode, plus all their derived forms. Something that it corrosive gnaws away at something else, as erosion gnaws away at the terra firma until it is much less firma. :)
In sum, the only words I could find in English that derive from Latin rodere directly or indirectly are
anticorrosion, corrodability, corrodable, corrode, corroded, corrodent,
corroder, corrodiary, corrodiating, corrodibility, corrodible, corrodier,
corroding, corrody, corrosibility, corrosible, corrosibleness, corrosion,
corrosive, corrosively, corrosiveness, corrosiving, corrosivity, erodable,
erode, eroded, erodent, eroding, erose, erosion, erosional, erosionist,
erosive, erosive, eroso-, incorrodible, incorrosive, rodent, rodential,
rodentially, rodentian, and — alas! — rodenticidal and rodenticide.
They’re all of them about gnawing, one way or the other.