Sex discrimination in the terms or conditions of employment, including but not limited to pay, has been illegal in the U.S. for more than half a century. An employee or prospective employee who's discriminated against can recover a money judgment against the offending company.
From the cursory treatment in the linked article, I can see two possibilities. It states:
Under the new law, companies that employ more than 25 people will have to prove to the government that they're paying men and women equally.
That might mean a reversal of the burden of proof. In the U.S., the person claiming illegal discrimination has the initial burden of presenting evidence to make it more likely than not that there was discrimination. Only then is the employer required to refute that evidence and/or show that there was a valid non-discriminatory reason for the challenged decision(s). The practical problems of the burden of proof are eased in many cases because (1) the complainant begins the process by going to the EEOC, which can bring its resources to bear, and (2) often, multiple individuals join together to bring a class action.
The statement that, in Iceland, the company "will have to prove" nondiscrimination might mean that every company has to present a case whether or not it's sued. A reversal of the burden of proof would matter in some cases but it's not exactly revolutionary. Remember that "proof" doesn't mean it's definitive. In this context, it just means more likely than not.
The other possibility I see is that it means equal pay, period, as opposed to the "equal pay for equal work" standard that Bernie invokes in his quoted comment. Much of the pay gap in the U.S. arises because cultural factors push women into occupations that pay less. Sometimes they pay less because of discrimination but sometimes it's because of market forces. If a law firm's lawyers are mostly men (reflecting the percentage of male graduates of the law schools from which it recruits) and its secretaries are mostly women, would Iceland require that pay scales be adjusted to provide for equal pay regardless of job title? That seems unrealistic.
I've seen this headline elsewhere but I still don't understand the "first country" reference.