"Reversed" Shapiro–Wilk The Sharipo-Wilk test, according to wikipedia, tests the null-hypothesis ($H_0$) "The population is normally distributed".
I am looking for a similar normality test with $H_0$ "The population is not normally distributed".
Having such a test, I want to calculate a $p$-value to reject $H_0$ at significance level $\alpha$ iff $p < \alpha$; proving that my population is normally distributed.
Please note that using Sharipo-Wilk test and accepting $H_0$ iff $p > \alpha$ is an incorrect approach since it literally means "we haven't enough evidence to prove that H0 doesn't hold".
Related threads - meaning of $p$-value, is normality testing useless?, but I can't see a solution to my problem.
The questions: Which test should I use? Is it implemented in R?
 A: 
I want to calculate a p-value to reject H0 at significance level α iff
  p<α; proving that my population is normally distributed.

The normal distribution arises when the data is generated by a series of additive iid events (see the quincunx image below). That means no feedbacks and no correlations, does that sound like the process that lead your data? If not, it is probably not normal. 
There is the off chance that type of process may be occurring in your case. The closest you can come to "proving" it is to collect enough data to rule out any other distributions that people can come up with (which is probably not practical). Another way is to deduce the normal distribution from some theory along with some other predictions. If the data is consistent with all of them and no one can think of another explanation then that would be good evidence in favor of the normal distribution.

https://en.wikipedia.org/wiki/Bean_machine
Now if you do not expect any specific distribution a priori it may still be reasonable to use the normal distribution to summarize the data, but recognize that this is essentially a choice out of ignorance (https://en.wikipedia.org/wiki/Principle_of_maximum_entropy). In this case you do not want to know whether the population is normally distributed, rather you want to know whether the normal distribution is a reasonable approximation for whatever your next step will be.
In that case you should provide your data (or generated data that is similar) along with a description of what you plan to do with it, then ask "In what ways may assuming normality in this case mislead me?"
A: You will never be able to "prove" a Normality assumption in your data. Only offer evidence against it as an assumption. The Shapiro-Wilk test is one way to do this and is used all the time to justify the Normality assumption. The reasoning is that you start off by assuming Normality. You then ask, does my data suggest I'm making a silly assumption? So you go ahead and test it with Shapiro-Wilk. If you fail to reject the null hypothesis then the data doesn't suggest you're making a silly assumption. 
Notice, people use this similar logic all the time in practice - not just in the context of the Shapiro-Wilk test. They want to use linear regression, look at a $Y, X$ scatterplot and see if linear regression is a silly idea. Or, they assume heteroscedasticity and plot error terms to see if this is a silly idea.
A: There is no such thing as a test that your data are normally distributed.  There are only tests that your data are not normally distributed.  Thus, there are tests like the Shapiro-Wilk where $H_0\!: \rm normal$ (there are many others), but no tests where the null is that the population is not normal and the alternative hypothesis is that the population is normal.  
All you can do is figure out what kind of deviation from normality you care about (e.g., skewness), and how big that deviation would have to be before it bothered you.  Then you could test to see if the deviation from perfect normality in your data was less than the critical amount.  For more information on the general idea it might help to read my answer here: Why do statisticians say a non-significant result means “you can't reject the null” as opposed to accepting the null hypothesis?
