This is an interesting question, and one that I can only really give an unsatisfactory answer to, as I've not looked into the primary sources for myself. Instead I'll be following the interpretations given by Michael Alpert, the foremost English-language scholar of the Republican army. If you want to look into the question in more depth, I can point you in the direction of some of the relevant Spanish-language resources at the end.

In Alpert's view, Vicente Rojo Lluch (Rojo hereafter) falls squarely among the 'geographically loyal' (leales geográficos) - that is, officers who would have served on either side but happened to find themselves in Republican territory in the early stages of the war. It's certainly true that Rojo had family in the Republican zone, and may well have felt compelled to stay loyal for their sake. However, I think Alpert is perhaps a bit harsh here, as it seems that Rojo had the opportunity to join the besieged officers in the Alcázar of Toledo, and he chose to remain loyal and tried instead to convince the rebels there to surrender (while interceding on behalf of their families in Madrid). According to his own writing, his chief concern during the civil war was to preserve the military hierarchy itself, evidently including the elected Republican government. Interestingly, at his eventual trial for 'military rebellion' - the paradoxical charge levelled at loyalists after the war - he was asked whether he had ever considered defecting to the Nationalist side. Rojo claimed to have wanted to, for 'sentimental' reasons, but saw his duty as lying with the Republican government. He was found guilty and sentenced to 30 years in prison, but pardoned soon after without serving any time.

Although you characterise him as a Catholic conservative above, it's worth distinguishing his social background from his politics. We might conclude that he was a somewhat more modern thinker than this background indicated (it was Rojo who translated various works on tank warfare for the Toledo military academy). But Rojo was not a member of any party; he had avoided politics completely before the war, a habit he tried to continue as an increasingly important Republican officer. Soviet reports from Spain, for instance, indicate that he remained completely disinterested in politics. In the context of the Republican Army though, this was actually a failing - it was an inherently politicised organisation, and ignoring politics essentially meant ignoring part of his job. It meant he became relatively detached from many of the concerns of his subordinates, and left him unable to resolve disputes, such as complaints that communists were gaining too much influence within the hierarchy.

I hope this goes at least some way towards answering your question. Rojo was not unique in staying loyal despite having no particular political connection to the Popular Front government (or separatist groups), although few others reached the same high rank. For many, this was simply a pragmatic choice - they and/or their families were on one side, and their safety depended on being seen to be loyal. Others, likely including Rojo, saw their duty as lying with the institutions of the government and army. To my mind, this does not indicate a particular respect for democracy, but rather a respect for process and hierarchy.

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Most of the above is from Michael Alpert, The Republican Army in the Spanish Civil War, 1936–1939 (Cambridge, 2013). If you want to read further, Rojo wrote three books about the war: ¡Alerta los pueblos! (Buenos Aires, 1939); Así fue la defensa de Madrid (Mexico City, 1967); España heroica (Buenos Aires, 1942). These might also be useful: Martínez Paricio, J., Aspizua Turrión, J., Ramón, J., Urbina, B. and Molina Benayas, J., eds., Los papeles del general Rojo, (Madrid, 1989) and Rojo, J. A., Vicente Rojo, retrato de un general republicano (Barcelona, 2006).

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