After Congress sends a copy of an approved bill to the President, he has 10 days, not counting Sundays, to act on it. If he signs it, the measure becomes law. If he fails to act within 10 days while Congress is in session, the legislation becomes law without his signature.

But if he fails to act within 10 days while Congress is in recess, as it is now, the measure dies. For the children's television bill, the 10 days expired at midnight Saturday, and the White House, without explanation, waited until the final hours to announce the decision. Throughout most of the day Saturday aides asked about the fate of the bill responded only by saying that the President was not expected to take action on any legislation that day.

One major opponent of the bill was Dennis R. Patrick, the chairman of the Federal Communications Commission, who said last month when the Senate passed it, ''The bill is both unnecessary and ill-advised.''

Mr. Patrick said it was unnecessary because the commission had pending proposed rules aimed at over-commercialization of children's television. A spokeswoman for the commission said that Mr. Patrick had no comment beyond that statement. Bush and Reagan Assailed

Senator Timothy E. Wirth, Democrat of Colorado, who had favored a measure even more stringent than the one passed, said after the Senate voted last month that the present system was not working. ''As a result of the F.C.C.'s neglect of this problem,'' he said, ''a troublesome situation has grown much worse.'' Representative Markey, who is chairman of the House Energy and Commerce Subcommittee on telecommunications and finance, said: ''President Reagan's action punctuated with an exclamation point the Federal Communications Commission's decade-long assault on children's television and exposed the hypocrisy of the Reagan Administration.''

While saying that children's television ''should not be and need not be a partisan issue,'' Mr. Markey sought to turn the veto to political advantage. ''Is this veto what Vice President Bush means when he pledges to be the education President?'' Mr. Markey asked. ''The Reagan-Bush team extolls family values and education but never misses an opportunity to take concrete action to deprive our nation's children.'' Constitutional Issue Raised

In announcing his action, President Reagan said: ''While I applaud efforts to increase the amount and quality of children's television programming, the Constitution simply does not empower the Federal Government to oversee the programming decisions of broadcasters in the manner provided by this bill.''