The simple answer to this is No. Once you die and a grant of probate is obtained to enable your executors to deal with your assets, your will becomes a public document. In contrast royal wills have historically been sealed for privacy reasons.

And understandably - after all, the publication of a will is the most astonishing invasion of our own, and our beneficaries' privacy. Anyone can access a document that shows what most would consider very private information -  the closest emotional ties that each of us make in our lives, including full names and sometimes addresses of those people: and by inference those we have omitted from benefit; and information about the size of our estate with the ensuing loss of confidentiality of our financial affairs and those who benefit. 

To make matters worse, there are no restrictions on who has access to this information. Anyone can search online and that information can be freely disseminated to a mass readership in minutes. With it comes the additional risk of harming the reputation of a person, should, as sometimes happens, the will contain a defamatory passage to explain the absence of a close relative.

Arguments to justify the release of wills seem weak when compared to the general access to probated wills. One is that interested parties may be alerted to the existence of an estate from which they should benefit. The second, that executors are thereby accountable; and thirdly, the information provides an informal census of wealth in the public's interest.

Until the law changes, and I believe it should, take care to limit the information you put in your will to the important issues of who is to benefit, and from what. Letters of wishes, which are never published, can deal with the difficult relationships, your funeral wishes, and other more personal information.