I think a big part of the reason Islam spread as quickly and as widely as it has was due to three factors. First, it is a heavily proselytizing religion, and with a very strong militant streak, meaning it sees nothing wrong with conquering in its name. Second, the deplorable state of Byzantine and Persian empires after 20+ years of open warfare, religious persecution of everyone who did not fall in line, and so on, leaving them both open to conquest. Third, conquered people often tend to take on the religious and the social traditions of the conquerors, providing that the conquerors settle in the conquered territories, and form a ruling class that is not quickly overthrown. The fact that in Islam, religion forms a basis for where in a society an individual falls, and conversion is often a way for social advancement of one and one's family, there is an added incentive for the conquered people to convert.
Now, let's do a quick check. If Islamic conquerors ignore Byzantium, where can they go? Without conquest, Islam remains a religion of a rather backward region of the Dark Ages world. There is no incentive for anyone to convert to it, such as trade links, being somewhat attached to a great and powerful civilization, etc etc (see Malay conversion to Islam in OTL as an example of a culture adopting Islam through trade and diplomatic links as opposed to being directly conquered). Much of Islam's eventual advancement was due to its conquest of formerly prosperous and advanced territories, whose culture and sophistication played very much into what is now called the "Golden Age of Islam". Most of those territories were Byzantine and Persian provinces at the time.
Let us figure out first why Arabs ignore Byzantium. If it is because they never managed to unify in first place, then it is safe to assume they are not powerful enough to take on either Byzantium or Persia. Which means Islam as a religion and as an ideology is pretty much stifled at birth, and remains a historical footnote. There are not many other places where the kind of an army the Arabs are likely to wield at the time could be as effective outside of Middle East or North Africa. Not to mention that the experience the Arab commanders gained from fighting Byzantine and Persian forces was sorely needed for the future conquests.
So, if the Arabs do not unite under the flag of Islam, then it remains a regional phenomenon, and not a major threat to either Byzantium, Persia, or anyone else. And without having wealthy, heavily populated, and advanced provinces with their human potential, Golden Age of Islam never starts in first place. Without the Golden Age of Islam, there is very little cultural incentive for anyone to convert to it, and without the resources or control of trade routes that eventual Arab conquests brought, there is very little economic incentive for anyone to convert to Islam. After all, conversion does not open new markets, does not bring new advancements, and does not make one a part of a greater cultural or social sphere in this scenario.
Now, let's presume that Islam DID unify Arabia. This postulates a question of just why would Byzantium be ignored. In order for this to be accomplished, perhaps an earlier, more overwhelming victory against Persia is required, so that the Byzantine forces are not essentially spent by the time of Yarmuk. Or, perhaps, an overwhelming defeat of the Arabs at their initial attempt at invasion would do the trick here. This means that the only places the Arabs can expand to are either Axum, or Persia.
If Axum is their next target, perhaps we can see Islam spreading further into Africa, but then, the question remains as to how long will it be possible to keep it from advancing into the Byzantine territories. If Persia, then consider that the Persians have already been thoroughly defeated by the Byzantines, and will probably fall just as they have IOTL.
For Islam, taking Persia is a better overall thing than taking on Axum. Axum, while relatively powerful for the region, is far from a powerhouse of technological and cultural advancement. Not to mention the generally good relations between the Axumite rulers and early Islamic leaders early on. There is just not much of an incentive to go there.
On the other hand, Persia is still relatively rich, quite advanced, and with quite a few people to form a strong economic and intellectual base. This may lead to Islamic civilization still being quite powerful and advanced down the line, although with more of an Eastern focus. And still it would be very likely to come into conflict with the Byzantines at one point or the other. Of course, given a few more decades, the Byzantines are more likely to recover and prove to be a "stopping block" for the Islamic expansion, which means that one opponent for Byzantium in the region will be simply replaced by another.
Now, what would be interesting to see would be how Islam would develop early on if it does not have any Hellenistic influences, which ignoring Byzantium would certainly cause to happen. Much of what we came to know as Islamic science and philosophy would not come to pass without conquest of formerly Roman territories that possessed foundations for it. Especially if Islam still unified Arabia, conquered Persia, and attempted to expand East, towards India. The implications of that would be quite interesting, especially since at that stage Islam would still be very much in its formative stages (as opposed to the later period, when most of its formative stages were through by the time it conquered enough of India)...