I don't think so.
First, she is American by birth and currently carries the title of "Duchess" not "Princess"....and yes, those things still matter in both English society and the British media. Also, just as Andrew was the more-affable and seemingly down-to-earth brother to Charles, Harry carries more charm than mystique, so I'm guessing Meghan will be considered a celebrity royal on the level of a Sarah Ferguson but never achieve the level of reverence of a Diana.
Next, the nature of the media and media-driven culture has changed greatly since the 1980s when Diana was portrayed as a real-life fairy-tale princess. I think today's culture is far more cynical and far-less likely to elevate Harry and Meghan to the same levels of adoration.