In Europe, you can be sued for not taking action on climate change. In the US, it’s the opposite.

ING joins a list of companies and other organizations that have been sued for not doing enough to tackle climate change in Europe and the U.K. On the flip side, companies and charitable institutions in the U.S. have been sued for focusing too aggressively on commitments to combat climate change, with nonprofits facing costly pushback in the courts.

The bank, which is being sued by Dutch environmental group Milieudefensie, has been one of the biggest players in the financing of commodities projects, including those related to fossil fuels. Milieudefensie says that as ING is one of the world’s largest banks, it bears responsibility for the impact its investments have on the climate.

ING said that Milieudefensie’s claims were “unrealistic and unreasonable," adding that it was confident in its climate approach. “We share the concern about the climate but differ in opinion on what action that requires," it said in a statement.

The suit follows a trial in the Netherlands last year that found oil giant Shell has a responsibility to combat climate change. That litigation was also brought by Milieudefensie, which is the Dutch wing of environmental group Friends of the Earth.

“Climate change is a direct threat to human rights, and the court has ruled that companies such as ING must take accountability for reducing their emissions," said Donald Pols, chief executive of Milieudefensie. ING must align its climate policies with the Paris Climate Agreement to do their part and help secure a sustainable future."

For ING, a ruling isn’t expected until next year, but if the case against the Dutch bank is successful it could open the doors to further legal action in Europe against fossil-fuel companies and the corporate actors that work with them, such as banks and asset managers.

The legal action is in contrast to the situation in the U.S., where banks and asset managers have been sued for focusing too aggressively on commitments to combat climate change and environmental nonprofits have faced costly pushback in the courts.

This month, the U.S. fossil-fuel industry secured a victory over environmental charity Greenpeace, which was ordered to pay hundreds of millions of dollars to an oil pipeline company after a jury found the charity liable for defamation and trespassing. Greenpeace has said it will appeal the decision.

In December, Texas Attorney General Ken Paxton said BlackRock and other asset managers were avoiding their fiduciary duty to shareholders by making climate-focused investments instead of investing in markets such as coal in their search for profitable returns.

In the U.S. regulatory sphere, the Securities and Exchange Commission paused its defense of proposed climate-disclosure rules under which large companies would report on their emissions, similar to regulations adopted in Europe and California. The SEC faced lawsuits saying the proposed rules were arbitrary and capricious and violated the First Amendment.

With lawsuits under way on both sides of the Atlantic, companies and charities that operate in both jurisdictions are being put in a bind. In the U.S., being sued for taking action on climate is becoming more likely, especially amid a Trump presidency. In Europe, litigation working in the opposite way is equally likely.

For many companies, the answer at the moment appears to be to keep their heads below the parapet to avoid criticism, and possibly costly litigation.

“Global corporates and financial institutions have to really think hard about what they’re doing, how they’re doing it, and what they’re saying about it," said Ken Rivlin, global co-head environment, climate, and regulatory law at A&O Shearman, a legal firm which operates in both Europe and the U.S., referring to actions on climate and sustainability. “It’s something that is top of mind in the C-suite with almost all of our clients right now. And there’s no easy answer right now."

However, some companies and NGOs in the U.S. are trying to push back.

In Texas, the American Sustainable Business Network is suing Attorney General Paxton and Glenn Hegar, comptroller of public accounts, over Texas Senate Bill 13, which bars state public entities from investing and contracting with companies on the basis of their actual or perceived political views regarding fossil fuels. The ASBN said this violated the first and 14th amendment rights to free speech and stopped investors from being able to choose where to put their money and how to spend it, including in renewable energy.

“I think most companies are going to do what makes sense for them financially," said David Levine, co-founder and president of ASBN. “We have to be vocal about this political and legal threat and we should not be intimidated out of doing the right thing for our businesses and our economy."

Greenpeace said the litigation it faced recently could set a precedent for further action. “Every step of the way, we’ve emphasized that these types of lawsuits — intended to silence and shut down critics—are part of a growing national attack on our First Amendment rights," the nonprofit said after the jury decision last week. “The truth is, ‘victory’ doesn’t just mean defeating this specific lawsuit. It means that no other organization or individual has to defend themselves against this type of attack."

How this plays out on either side of the Atlantic isn’t clear-cut. The European Commission—the executive arm of the European Union—recently watered down its flagship climate accounting policies amid pushback from countries within the bloc and from companies that said the reporting requirements were too arduous and costly, putting them at a disadvantage to those in the U.S. and Asia.

A trial last year in Switzerland may have set a precedent. The European Court of Human Rights ruled in favor of a group of elderly Swiss women who argued that their government wasn’t doing enough to fight climate change, putting them at risk of death from heat waves. In effect, it ruled that governments have a responsibility to protect citizens against the effects of climate change.

“You may see litigation as your best way of having an impact," A&O Shearman’s Rivlin said. “And even if you don’t win the case, you’re bringing press to it. I think a similar kind of approach is going to drive how attorneys approach things. It’s just that depending on if it’s a red state or blue state, the approach is going to be different."

Write to Yusuf Khan at yusuf.khan@wsj.com