Shares of Netflix rose 1.8 percent Thursday, reaching a market value of $152.6 billion, according to FactSet. Disney's stock fell 1 percent and now has a market value of $151.8 billion.
Netflix closed off its highs, up 1.3 percent with a market value of $151.8 billion. Disney trimmed its losses to close 0.76 percent lower, for a market capitalization of $152.2 billion.
Market values of Netflix, Walt Disney and Comcast
Disney is getting more aggressive in the streaming online video market to compete against Netflix. The media company rolled out ESPN+ last month, its first-ever direct-to-consumer service, which includes live sports and other programming. It costs $4.99 per month or $49.99 for the year. The company also will launch a Disney-branded streaming service next year.
But one Wall Street analyst thinks Netflix is just getting started in its international subscriber growth.
"We believe Netflix still has a considerable opportunity ahead if it can achieve reasonable penetration levels internationally," Bank of America Merrill Lynch analyst Nat Schindler said in a note to clients last week. "Netflix will face varying levels of competition, regulation and economic conditions in each individual market it participates in, but its content scale should allow it to become the dominant streaming player in virtually all markets."
Schindler predicts the company's subscriber base can grow 8 percent annually through 2030 to 360 million members. Netflix revealed its current membership level was 125 million subscribers at the end of its first quarter.
Netflix is the best-performing stock in the S&P 500 so far this year. Its shares are up nearly 82 percent through Thursday, while Disney's stock is down about 5 percent.
Disclosure: Comcast is the owner of NBCUniversal, parent company of CNBC and CNBC.com.