So, broadly speaking the premise here is a little off. The International Brigades were officially withdrawn from Republican Spain about six months before the end of the conflict. While this process was slow, and many didn't really have anywhere to go because their home government wanted nothing to do with them (or in the Germans' case was too interested in their whereabouts for comfort), it was a much easier process for those who had come from democratic states. While some such governments, especially Canada for some reason, did put up some barriers or refused to given volunteers access to diplomatic representation or assistance, most English-speaking volunteers were home by the end of 1938. Even for those that remained in Spain, however, the foreign volunteers had been based in Catalonia - alongside hundreds of thousands of Republican soldiers and civilians, the remaining volunteers chose to cross the French frontier, and were interned in makeshift French camps instead of awaiting Francoist justice. Again, Americans were better off than the norm for people in this situation in that they had a government willing to put some effort into getting them home (as well as allies at home to push for their release and provide funds). Not all were so lucky - many were still interned in Fench camps over a year later in June 1940, when France suddenly became a rather unhealthy place for large groups of people with known anti-fascist sympathies.

So, your American likely avoided the specific fate you mention. However, Nationalist forces did capture foreign volunteers in battle, so we can say a bit more about how they treated American prisoners. In some cases - though it's difficult to say exactly how often - foreigners were summarily executed when taken prisoner. In one account from March 1938, a Spanish conscript serving alongside the foreigners recalled their captors separating out the (American) foreigners from the Spaniards by asking them questions, in Spanish, about their homes. The foreigners were marched off together, and the remaining Spaniards eventually heard gunfire in the distance, though whether these shots in fact came from the execution of the captive Americans has never been verified. I personally don’t doubt that execution, particularly in the early months of the war, was the common fate of most international captives. Tellingly, Franco himself issued an army-wide command in April 1937 that foreigners should not be shot, ordering that:

[Foreigners] lives be respected so that they can be repatriated to their country of origin, since a large number of those enlisted in the International Brigades want to defect if they know that their lives will be safe.

The necessity of issuing such an order is, to me at least, a strong indication that up to that point, executions of captured foreign fighters had been common practice. I also doubt that this order ended the practice entirely: the lack of foreign captives taken in a number of major battles is, I think, an indication that summary executions were fairly common, particularly for smaller groups of prisoners or individuals. The only British prisoner, Robert Beggs, taken at the Battle of Brunete, for instance, appears to have survived by convincing his captors he was seeking to defect having been deceived by the tricksy communists into coming to Spain – he was however a well-known member of the Communist Party of Great Britain in Glasgow...

For those that were captured alive, the process was not all that different to what Spanish prisoners of war experienced. Most ended up in a prisoner of war camp, where the regimes were generally harsh, conditions poor and violent treatment at the hands of the guards common. Yet although the prisoners themselves made the comparison with German concentration camps, it’s worth noting that the conditions were not so bad as that: there were few deaths among the prisoners, and they were often able to receive packages and money from home. Some governments were able to make representations on behalf of their nationals, although there were in practice limits to such diplomatic efforts to shield prisoners from harm. British diplomats, for instance, were somewhat too willing to trust to the ‘gentlemanly’ commanding officers’ good intentions, and were predisposed to disbelieve the generally left-wing, working-class volunteers’ claims of mistreatment. Such mistreatment was certainly a reality though – the aforementioned Beggs learned this the hard way, when he ran afoul of a guard sergeant, who beat him so severely Beggs ended up in the camp hospital for weeks. This incident, among others, earned the sergeant the nickname of ‘Sticky’ among the English-speaking prisoners. This was less a remark on his adhesive qualities, but reference to his apparent fondness for carrying around and using a large stick to beat the prisoners with.

For volunteers from fascist countries such as Germany, the reach of the secret police was an added danger. At the camp of San Pedro de Cardeña, where just about all foreign prisoners had been concentrated by early 1938, there was a ‘Gestapo’ presence that paid particular attention to the German prisoners, subjecting them to especial interrogations. All prisoners, however, were interviewed extensively. Aside from the usual pursuit of intelligence, there were some truly bizarre efforts to analyse the captives. Many prisoners of war were subjected to detailed physical and psychological examination by the Spanish psychiatrist Antonio Vallejo Nágera, who sought to discover the psychological causes of Marxism. Vallejo Nágera subjected groups of Portuguese, American and British prisoners to an array of strange and often humiliating tests, seeking to prove the latent inferiority of ‘Reds’ and connect their deviancy with their lack of exposure to Christian religion and traditional values. Interestingly, Vallejo Nágera’s experimental results are the best source of some key demographic data about the foreign volunteers, as the International Brigades themselves neglected to record volunteers’ religion or marital status.

The volunteers made efforts to mitigate their circumstances. Senior communists managed to form secret committees in an effort to organise the prisoners, though their efforts were hampered by the belief that the Nationalists would execute anyone suspected of being a Political Commissar or officer. It also didn’t help that many of the volunteers were extremely demoralised, even going so far as ‘openly declaring their preference for Fascism’ and ‘applaud[ing] the Italian officers.’ Tensions also came to the fore in the various ‘sharp divisions’ that cropped up throughout their imprisonment, on issues as varied as whether they should agree to shout ‘Franco’ on parade, or if they should campaign for better food. While the organisers boasted of their efforts to set up a camp ‘university’ to give lectures and teach inmates new skills, privately they admitted that it ‘had to be discontinued due to the activities of informers and the attitudes of the officers.’ Despite this, they were able to establish a camp fund that was used to buy luxuries such as extra food and cigarettes.

For those with governments able to intercede on their behalf, eventual repatriation was the most common fate for those in such camps. The first large group of British prisoners, taken at Jarama in February 1937, were released by May the same year, though not before they had gone through what amounted to a ‘show trial’, and handed down sentences ranging from execution to varying spells of imprisonment. While these were obviously not enforced, they were required to swear an oath not to return to Spain and take up arms again. Several chose to ignore this promise, and re-enlisted in the International Brigades. One, Jimmy Rutherford, had the misfortune of being captured again in March 1938. He was eventually recognised by one of his interrogators, and sentenced to death – a sentence which this time was carried out swiftly. British official protests at the execution were muted – as the Foreign Office privately acknowledged, the Nationalists had a fairly watertight case. Yet Rutherford’s fate was unusual, and most English-speaking prisoners had been sent home by the end of 1938, with the final few repatriated by February 1939. Those with no nationality, or were from countries which didn’t want them back, could languish for far longer in Franco’s camps – some were still imprisoned five years after the end of the civil war in 1944. So, providing that your American volunteer survived the initial capture, they would have had a tough few months, but would likely have survived to tell the tale thanks to the rlative influence their government was able to wield.