HONG KONG — Samsung, the giant maker of products as varied as cargo ships and smartphones, is one of the driving forces of the South Korean economy. Now, the fate of its leadership could be tied to the sprawling investigation of a corruption scandal involving the nation’s president.

Prosecutors on Monday called for the arrest of the company’s heir-apparent, Lee Jae-yong. Prosecutors contend that Mr. Lee bribed President Park Geun-hye and one of her confidants in exchange for political favors.

The charges were leveled at a tough time for the company, which was struggling in some of its core businesses while trying to break from its past and forge a new management culture. Samsung’s Galaxy Note 7, intended to showcase its ability to innovate and challenge Apple for global smartphone supremacy, instead showed a propensity to burst into flames, and was recalled. Samsung has been without a chairman for two years, after the patriarch had a heart attack.

The crisis is particularly difficult not only for the company, but for South Korea’s relatively young democracy. One of the country’s most important companies might be wounded, and the nation’s leadership has been thrown into disarray.

Suave and mellow, Mr. Lee has often been portrayed as the right leader for a vast, stodgy business empire needing to refocus.

The bribery crisis has dominated the news for days in South Korea, where Samsung and other huge “chaebols,” or family-controlled industrial empires, are embedded in politics and in the national identity. Samsung alone accounts for about 20 percent of South Korea’s exports.

The country’s largest chaebols have faced legal battles and convictions in recent years, but these episodes often end with the industrialists receiving pardons.

Mr. Lee might not be arrested at all. That will be decided this week.

Prosecutors say Samsung made payments in exchange for a decision by the government-controlled National Pension Service to support a contentious 2015 merger of two Samsung affiliates. Analysts say the merger helped Mr. Lee inherit control of Samsung from his father.

In a statement on Monday, Samsung denied any bribery or making “improper requests related to the merger of Samsung affiliates or the leadership transition.” But the move by prosecutors casts Mr. Lee in a different light, as a leader similar to his father, who was convicted of white-collar crimes but pardoned twice. It also could raise questions about Mr. Lee’s pledge to make Samsung more transparent and more responsive to shareholders.

South Korea’s president, Ms. Park, has denied wrongdoing.

Mr. Lee, who goes by Jay Y. Lee in the West, is part of the third generation of the Lee family to lead the conglomerate. While his father was widely seen as old-fashioned, Mr. Lee has been painted by Samsung as equally comfortable in Silicon Valley and in the boardroom, as a leader who could open up the clannish business and reinvigorate Samsung’s flagging reputation for innovation in consumer technology. Samsung is the world’s biggest maker of cellphones.

“Beyond the arrest itself, this is going to be a big blow to the narrative they’ve been building,” said Geoffrey Cain, the author of a coming book on Samsung. “It’s hard to convince shareholders and partners they are a hip Silicon Valley-style company when these charges show them to be a company run like a feudal dynasty.”

Samsung is in effect a collection of widely disparate companies, each with an autonomous, professional management team. Samsung Electronics, the most important of those companies, is broken into high-functioning business units. Because of that, some observers say that the company could continue to run smoothly without Mr. Lee.

Important matters that require Mr. Lee’s imprimatur could be delayed amid the distractions, said Mark Newman, an analyst at Sanford C. Bernstein and Company. “I think that’s the thing you need to worry about: big decisions,” he said.

If executives lower in the hierarchy are implicated, day-to-day operations could become complicated. According to South Korean news reports, other top company leaders have been questioned.

Mr. Lee’s role at Samsung is not necessarily imperiled. South Korea has a long history of pardoning the leaders of its largest family-run companies. Because of Samsung’s size, past efforts to penalize the company or its executives have been met with worries about the effect on South Korea’s economy.

But Mr. Lee’s problems could influence the company’s strategic decisions at a time of major challenges, such as how to bounce back from the debacle over its Galaxy Note 7. A spate of fires last year forced Samsung to recall and then cease production of the phone, tarnishing the company’s name and making it the object of airline safety announcements and jokes on American late-night talk shows.

Analysts highlighted two executives who run large parts of the company’s operations who they said could take on more responsibility. One is Kwon Oh-hyun, who helped Samsung become a leader in memory and semiconductor operations. The other, Choi Gee-sung, was crucial in building the company’s television unit, which eclipsed Sony’s a decade ago.

Mr. Lee, 48, is vice chairman of Samsung, but he has been widely considered the company’s de facto leader since his father, Lee Kun-hee, had a heart attack in 2014. The elder Mr. Lee led Samsung from its roots as a components supplier to a maker of consumer products, establishing a global brand and shedding a reputation for cheap goods. In one example of his drive, two decades ago he gathered 150,000 Samsung phones that failed quality tests and set them on fire.

Under the younger Mr. Lee, Samsung has focused again on components, a sector that has benefited from companies wanting to add features to their phones. Late last year, the company struck an $8 billion deal to buy Harman International to make smart components for cars.

But Mr. Lee has also pledged to make changes to ensure that Samsung’s corporate governance meets international standards. As a family-controlled group with dense links among its units, the corporation has a structure that has drawn criticism from some investors.

Samsung has also pressed for cultural changes to foster innovation and a start-up-style environment, and Mr. Lee has been portrayed as a part of that shift. At Samsung, he is credited with negotiating an important deal with Steve Jobs to help make the Apple iPod.

“Start-up Samsung,” an initiative announced last year, sought to relax what many described as a strict corporate culture. Announced with a coordinated pledge by top executives, the push was derided by some in the technology news media as evidence of Samsung’s top-down structure.

Despite such efforts, Samsung has not followed up with a next act in the wake of the Galaxy Note 7 debacle, and prominent projects like creating an operating system have largely been considered failures.