As far as I know, it's not bullying since I've never seen or heard of anyone going further than asking him. If he doesn't want to admit it, yes, it's his business. It's a shame though - a lot of older people are homophobic and see gay people as living some sort of seedy, unwholesome lifestyle. If Cliff, the symbol of smiley-faced, clean-living normality were to come out, he could probably do a lot to help change that perception. I think you'll find most of his fans are in denial.
Also, if I were his alleged partner (perish the thought!) I don't think I'd be too happy about my boyfriend/husband hiding me out of public view, refusing to be seen with me at events and refusing to acknowledge/actively denying a long term relationship.
To be honest I think Cliff's pretty spineless. Too scared to change style either presentation-wise or musically, too scared/dull to veer out of the middle of the road at any point during his 50 years of recording, and as all evidence suggests, too scared to admit who he is. If he wants to live a nice safe life where he ploughs on with the same song and dance (figuratively and literally) for half a century, fine, but I don't find anything to admire in that.