Why couldn't Merry ride his own horse to Gondor?

[Kalimac]

I'd say that "able to ride a horse" implies that Bandobras actually did it (he's also mentioned in The Hobbit, as being "so huge - for a hobbit - that he could ride a horse.") It would be a strange statement to make if he hadn't actually done so - it doesn't have that obviously-metaphorical feel that something like "He was so huge he could chop down a forest in single day" would have.

As for Merry - just because he's now big enough to ride a horse (he's about the size of some of the smaller jockeys) doesn't mean that he's automatically going to know how to do it, and I'd imagine that riding his pony earlier was a somewhat different experience. To pull him off the pony and put him onto a war-horse would not necessarily produce great results. Furthermore, like anyone who's just had a growth spurt (and he and Pippin have had substantial ones) Merry's probably going to be a bit clumsy in his new body at first; more liable to knock into things or mishandle them. It isn't the time for him to be learning how to handle a new animal in a life-or-death situation, and there's no time to train him.

Now from Theoden's point of view, he's probably just looking at Merry and thinking "Too short, too inexperienced in fighting." It may not cross his mind that technically Merry is big enough to ride a horse, but I doubt that would change his mind if it did. The above objections would probably be enough for him, most notably the one about having no time to teach him.