Short answer: Probably not.
Let's first keep in mind that very few split-brain patients have ever been studied. Add to this that inner speech is not a well studied area in general. As the resulting Venn diagram suggests, there is next to nothing known about inner speech in split-brain patients.
Second, remember that one of the key reasons why inner speech is problematic to investigate is because up until recently, our main data source has been self-reports, considered notoriously unreliable as a methodology. This is especially exacerbated in patients with awareness deficiencies such as split-brain.
Having said that...
Split-brain patients in general report no significant post-operative changes in their cognitive function, sense of self, or conscious experience. Internal monologue remains intact, and appears to function as normal. However testing under lab conditions demonstrates that subjects do in fact experience significant changes in awareness - effectively neglecting half of their perceptual experience.
If self-reports of awareness are so significantly inaccurate, then what are we to make of split-brain patients' self-evaluation of conscious experience and other cognitive functions? The truth is, we just don't know.
Relatively recently, new techniques for studying inner speech have been developed, such as fMRI imaging of subjects instructed to or prevented from engaging in self-reflection and internal monologue. Such studies suggest that inner speech is primarily localized to Broca's area (LIFG, on the left side of the brain), just as is outer speech (McGuire, 1996; Morin & Hamper, 2012). These findings are consistent with other evidence that inner speech is a form of silent (simulated) outer speech, and thus uses the same physical mechanism.
In the split-brain paradigm experiments, subjects are able to articulate stimuli presented to their right field of view (left hemisphere, where Broca's area is), but unable to do so for stimuli presented to their left side (right hemisphere, where language is lacking). To communicate with the right hemisphere, researchers have subjects use their left hand to gesture responses in some way.
These experiments were instrumental in localizing language function in the brain, and they demonstrated that language capabilities are largely lateralized in the left hemisphere. However, the right hemisphere has sufficient language understanding capabilities to gesture meaningful responses without the left hemisphere's awareness.
So although it's tempting to suggest that inner speech, localized in Broca's area on the left side of the brain, should be unaffected by corpus callosotomy, as reported by split-brain patients, the language capabilities of the right hemisphere leave open the possibility that a separate experience may exist there, potentially implying some reduction in function of (outer and inner) speech in the left side.
And indeed, while language function is highly lateralized in 90% of right-handed people, function is more evenly distributed amongst the hemispheres in other cases. By some stroke of luck, this was the case in a particularly interesting split-brain patient by the name of Paul S.
Remarkably, in the experiments conducted with Paul, he was able to use his left hand to spell out single word answers using Scrabble pieces to express what his right hemisphere was thinking (LeDoux et al, 1977). As with other split-brain subjects, his right hemisphere had a quite different experience from his left, but unlike other patients, Paul was able to demonstrate a slightly different personality residing in his right brain, with different opinions, preferences, and decisions.
Could this mean that Paul S. had a second, independent internal monologue in his right hemisphere that the left hemisphere was unaware of? The dual-consciousness interpretation of this case study has been the subject of much debate, but most scholars suspect that this is unlikely: Language function is simply too rudimentary in the right hemisphere, even in this remarkable case, to plausibly support this hypothesis.