A drumbeat of terror attacks in Western Europe is forcing a political reckoning on the continent’s two main powers, Germany and France.

In recent days, far-right parties and some mainstream conservatives in both countries have turned up pressure on the governments of Chancellor Angela Merkel and President François Hollande to abandon the incremental approach they have taken in fending off attacks by Islamic State and other radical groups.

Alternative for Germany, an upstart anti-immigrant party, proposed a ban on Muslim immigration, seizing on the attacks to build on its gains in recent state elections. In France, right-wing lawmakers rejected government demands for national unity, depriving Mr. Hollande of the political cover he has relied on to take a measured response in the wake of past attacks.

“I’m hearing a lot of words, seeing candles and calls for peace and fraternity, but in the global strategy of the government I’m seeing nothing,” said Marion Maréchal Le Pen, a National Front lawmaker and niece of Marine Le Pen, the nativist party’s leader. The 26-year-old said she would enlist as a military reservist and called on “young French patriots” to join her in reinforcing the country’s stretched security forces.

Both Mr. Hollande and Ms. Merkel have hewed to an approach rooted in restraint and pragmatism, putting more police on the streets and pledging better intelligence cooperation with European and Middle Eastern neighbors. Ms. Merkel hasn’t commented publicly on the first Islamist suicide attack to touch German soil.

That careful response, forged after the January 2015 assault on Charlie Hebdo’s newsroom in Paris, has helped foil some plots while seeking to avoid social conflict in historically Christian countries hosting the continent’s two biggest Muslim populations.

As acts of gruesome violence have started coming within days of one another, however, calls for unity are being drowned out by those demanding a dramatic shift to hard-line proposals, from a crackdown on immigration to restrictions on civil liberties.

Hours after they slaying of a Roman Catholic priest Tuesday in Normandy, former President Nicolas Sarkozy unleashed a blistering critique of Mr. Hollande’s government. He called for “profound change” including putting electronic monitoring bracelets on people deemed at risk of radicalization, or even placing them in detention centers.

“We need to Israelize our security conditions,” said Hervé Morin, president of the Normandy region and a former defense minister under Mr. Sarkozy.

On Wednesday Mr. Hollande strained to rally the nation, announcing the deployment of 23,500 police, soldiers and reservists to bolster security at 56 high-security events across the country this summer. He also called a snap meeting of religious leaders at the Élysée Palace.

Interior Minister Bernard Cazeneuve rejected many of the hard-line proposals as “an abandonment of the Republic and its principles.”

“You can’t protect the rule of law by sidestepping it,” he said.

Tuesday’s church attack in northern France has crystallized long-held fears in European intelligence services that the continent risks sliding towards a clash of civilizations, coming on the heels of the truck assault in Nice, France, that killed 84 people and a suicide bomb blast this week outside a music concert in Ansbach, Germany, both claimed by Islamic State.

In a closed-door hearing of a French parliamentary committee in May, Patrick Calvar, the director of the country’s domestic intelligence service, warned of a “confrontation between the extreme right and the Muslim world—not the Islamists, but the whole Muslim world,” according to a transcript released earlier this month.

“That’s what worries me when I talk with my European colleagues,” Mr. Calvar told the committee. “At some point, we’ll have to redirect resources to work on other kinds of extremist groups, because a confrontation is inevitable.”

In Germany, two terrorist attacks and a deadly shooting spree not linked to terror—all within a week—have shattered the sense of security in a country that had largely managed to avoided such attacks in recent years. While Germany has Europe’s largest Muslim population, it had escaped the kinds of high-profile Islamist attacks that have rattled neighboring Belgium and France.

“I used to love to come to Berlin to go shopping,” said Brigitte Zeibik, a 52-year-old justice-sector employee from Dresden who was passing through Berlin’s central train station on Wednesday. “But today, I honestly would have preferred to avoid it. I look around all the time.”

No one but the perpetrators died in Germany’s two Islamist attacks—an ax assault in Würzburg that injured five and a suicide bombing in Ansbach that hurt 15.

But the fact that attacks were both carried out by recently arrived asylum applicants—one who said he was from Afghanistan and one from Syria—cast a renewed spotlight on Ms. Merkel’s welcome of hundreds of thousands of refugees last year and her refusal to close the German border to migrants.

“For security reasons, we cannot afford to allow even more Muslims to immigrate to Germany uncontrolled,” said Alexander Gauland, deputy chairman of Alternative for Germany. “Ms. Merkel is ignoring the danger of terrorism and is thus isolating herself ever more.”

The concentration of the violence in Bavaria has also sown tensions between Ms. Merkel and a key governing coalition partner, the Bavarian Christian Social Union, or CSU.

CSU members in Bavaria’s state government have called for tougher background checks on new arrivals and a more aggressive stance on deportations.

“We have a very new dimension of terror, that is Islamist-shaped terrorism,” Bavarian Premier Horst Seehofer said Tuesday. “We must deal with this new challenge in Bavaria and in Germany as intensively as possible.”

The stakes are high. While Ms. Merkel has dominated Germany’s political scene for more than a decade, the three-year-old Alternative for Germany, also known as the AfD, has already become the most successful right-wing populist party in the country’s postwar history. A strong result in general elections next year could give the party its first-ever seats in national parliament and complicate Ms. Merkel’s path to a fourth term if she decides to run again.

Worries over migration and terrorism could give the AfD a chance to establish itself in Germany’s political landscape—a first in a country that had long seemed immune to nationalist populism because of its Nazi past.

“The hurdles are higher in Germany,” said Oskar Niedermayer, a political scientist at the Free University of Berlin. “But they are not insurmountable.”