No, the wage gap has been empirically shown to be real. Surely, much of the raw difference (“women earn only 79 cents for every dollar earned by men,” as Obama said) can be explained by occupation, number of hours worked, or experience. More specifically:
A greater percentage of women than men tend to work part-time. Part-time work tends to pay less than full-time work.
A greater percentage of women than men tend to leave the labor force for child birth, child care and elder care. Some of the wage gap is explained by the percentage of women who were not in the labor force during previous years, the age of women, and the number of children in the home.
Women, especially working mothers, tend to value “family friendly” workplace policies more than men. Some of the wage gap is explained by industry and occupation, particularly, the percentage of women who work in the industry and occupation.
So, to find the adjusted pay gap, i.e. the real pay difference between men and women of similar abilities/experience, working the same hours, and at the same jobs, we can use statistical techniques to control for these cofactors.
There have been several studies that have done just this. For example, a study commissioned by the Bush Administration’s Department of Labor found that, when controlling for the above factors, the pay gap is somewhere between 4.8%-7.1%. That is, they found that women are paid about 5-7% less than men who work in the same jobs, for the same hours, and have the same experience/qualifications.
There are observable differences in the attributes of men and women that account for most of the wage gap. Statistical analysis that includes those variables has produced results that collectively account for between 65.1 and 76.4 percent of a raw gender wage gap of 20.4 percent, and thereby leave an adjusted gender wage gap that is between 4.8 and 7.1 percent.
https://www.shrm.org/hr-today/public-policy/hr-public-policy-issues/Documents/Gender%20Wage%20Gap%20Final%20Report.pdf
https://www.bls.gov/opub/reports/womens-earnings/archive/womensearnings_2009.pdf
On the more progressive end of the political spectrum is studies commissioned by women’s groups, such as AAUW. They found that, after controlling for the pay difference between subjects, the pay gap is ~7%. That is, women earn 7% less than men with the same degrees one year after graduating from college. This still falls in line with conservative range estimates given by CONSAD, and in fact further reinforces the findings of a real pay gap within this range.
One might expect that when you compare men and women with the same major, who attended the same type of institution and worked the same hours in the same job in the same economic sector, the pay gap would disappear. But this is not what our analysis shows. Our regression analysis finds that just over one-third of the pay gap cannot be explained by any of these factors and appears AAUW 21 to be attributable to gender alone. That is, after we controlled for all the factors included in our analysis that we found to affect earnings, college educated women working full time earned an unexplained 7 percent less than their male peers did one year out of college.
http://www.aauw.org/files/2013/02/graduating-to-a-pay-gap-the-earnings-of-women-and-men-one-year-after-college-graduation.pdf
All the evidence, therefore, points to a pay gap of about 5-7% between men and women working under equal circumstances with the same qualification. Whether that is a significant number is apparently a matter of political dispute (I would argue that this range is more than simply “statistical noise”), but the fact remains that there is a real wage gap in which women are paid less.