By Daisuke Wakabayashi 

After a series of blockbuster earnings that blew past even the most optimistic of Wall Street expectations, Apple Inc. felt the sting of falling short of its own high bar.

Apple said Tuesday its profit surged 38%, aided again by strong demand for the company's latest iPhones and robust growth in China where sales more than doubled.

But while Apple sold 35% more iPhones in the fiscal third quarter compared with a year earlier, those sales missed some analysts' estimates. Apple also indicated its revenue in the current quarter could come in below Wall Street projections.

The iPhone is Apple's most important product, accounting for nearly two-thirds of Apple's revenue in the quarter ended June 27 versus less than half three years ago. Any signs that iPhone growth is reaching a peak is a major cause of concern for investors.

In an interview with The Wall Street Journal, Apple Chief Executive Tim Cook said he has heard repeatedly that the company can't sustain its growth rates, but it has proved doubters wrong in the past.

"I refuse to accept that type of thinking," Mr. Cook said. "I don't see the ceiling being very close."

Within minutes of the earnings report, Apple's shares fell as much as 7% in after-hours trading, erasing about $60 billion in market value.

Strong sales of the larger-screen iPhone 6 and 6 Plus are driving Apple to record earnings, while defying a sales-growth slowdown in the smartphone industry. Apple has pried open the door to largely untapped markets like China and enticed consumers to switch from smartphones running Google Inc.'s Android operating system.

The fiscal third quarter is traditionally weaker for iPhones as consumers hold off on purchasing new models until the fall when Apple typically introduces them. Analysts noted that iPhone unit sales of 47.5 million fell by about 23% from its fiscal second quarter, a steeper rate of decline than the previous two years when quarter-on-quarter sales fell by 19% and 17% respectively.

Apple said part of the shortfall was the result of it lowering iPhone inventory by 600,000 units during the quarter, a sign that it sold more phones than it manufactured.

Abhey Lamba, a senior technology analyst at Mizuho Securities, said iPhone sales falling short of some analysts' estimates hurt the bullish view of some investors that the iPhone 6 and 6 Plus could defy seasonal slowdowns the iPhone had experienced in the past.

"It deals a blow to that thesis," Mr. Lamba said.

The iPhone now overshadows the rest of Apple's businesses. While Apple has refreshed several product lines in the past year and introduced an entirely new category with a smartwatch, the iPhone accounts for an increasingly larger share of Apple's revenue.

On a call with analysts, Mr. Cook brushed off a question as to whether Apple needs to move beyond its reliance on the iPhone. "We think the phone has a lot of legs to it," he said.

In the latest quarter, iPhone sales made up about 63% of Apple's overall sales, compared with 53% in the year ago period and less than half three years ago. Apple has benefited from selling its iPhones at high prices at a time when smartphone prices are sliding. The average selling price of iPhones rose by more than $100 to $662.42.

The latest iPhones are especially popular in China. In greater China--defined by Apple as China plus Hong Kong and Taiwan--revenue more than doubled to $13.23 billion. For the iPhone, sales rose 87% in greater China compared with 5% growth in the overall market, according to Apple.

Mr. Cook said the company had the highest rate of switchers from Android phones ever during the quarter. "The gap is widening between us and our competitors," he said.

It isn't yet clear whether Apple's sales will greatly benefit from the Apple Watch, the company's first all-new hardware product since it introduced the iPad in 2010. Apple didn't provide a breakdown of the Watch's sales, which began during the June quarter, lumping the product's sales in with the iPod, Apple TV and Beats accessories in the "other products" category. Sales of that segment rose 49% to $2.64 billion.

Mr. Cook said the Watch's sales beat the company's own internal projection, although he didn't provide those estimates. He said the "sell-through" of the Watch was better than the iPad and iPhone at the same period of time. He also noted the Watch is still only available in 680 retail locations, or less than 1% of the locations where the iPhone is sold.

Apple's profit in the quarter rose to $10.7 billion from $7.74 billion in the year-ago period. Revenue jumped 33% to $49.61 billion.

Gross margin--a closely watched measure of profitability reflecting the percentage of revenue that remains after manufacturing costs--was 39.7%, above its estimated range of 38.5% to 39.5%.

Apple's forecast for revenue in the current quarter came in slightly below Wall Street's expectations. For the current quarter ending September, Apple said it expects a gross margin of between 38.5% and 39.5%. It sees revenue coming in between $49 billion and $51 billion. Analysts had expected revenue of $51.13 billion, according to a consensus of estimates compiled by Thomson Reuters.

In contrast to the iPhone's success, iPad sales continued to slump. Apple's tablet sales fell about 18% in unit terms, marking a sixth-straight quarter of year-over-year declines. The iPad and tablet computers, in general, are facing an existential crisis: They aren't quite as essential as the smartphone but not quite as functional as a notebook computer. Moreover, early iPad users don't see a huge reason to upgrade to more recent models.

Apple has struck deals with International Business Machines Corp. and other companies to position the iPad as a device for the workplace. So far, those efforts have done little to stem the iPad's slide.

Another bright spot is the Mac business. Apple said Mac sales rose 9% in units, compared with a 9.5% decline in shipments for PCs globally in the second quarter, according to research firm Gartner. While still one of the smaller players in terms of shipments, Apple has steadily gained market share on competitors.

Write to Daisuke Wakabayashi at Daisuke.Wakabayashi@wsj.com

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