First, what is the difference between rhotic and non-rhotic English? The answer here is the pronunciation of the /r/ sound at the end of words that we British English speakers tend to drop. Non-rhotic speakers tend to pronounce the word car as /kə/ where a rhotic speaker would pronounce it /kär/. Although rhoticity is usually associated with region (US accents tend to be rhotic while UK accents tend to be non-rhotic), there are some US accents which are non-rhotic.
The thing to note is that non-rhotic English does pronounce the /r/ when the next word starts with a vowel. In the sentence the car is white, the /r/ sound seems to start the next syllable /kə rɪz/.
What does this imply? If we were to switch to a phonemic alphabet, we would need to decide on one way of speaking. If we choose to standardize rhoticity, for example, then all non-rhotic languages would be disadvantaged as they would need to be educated to speak the way the way rhotic speakers do in order for their speech to match the spelling.
So I guess that the solution is to create a "standard English". If we were to create a superior form of our language which uses a phonemic alphabet and thereby standardizes the way that we speak, we could lead by example. In the same way that received pronunciation is held as a superior form of British English and is therefore preferred by broadcasters such as the BBC, mediums such as television and the internet could affect this change while regional accents are still maintained.
Of course there is the issue of whether any change is necessary at all. Perhaps as technology makes international communication more common, we will naturally standardize the way that we speak without the effort and expense of educating anyone at all. Just as the world is currently going through a process of natural selection with our languages (it has been said that the world loses 1 language every 14 days), so too will happen with accents. After all, the purpose of accents is to identify with the community in which you live and we now live in a global community.