Kenyatta asked journalists on Saturday to report more responsibly, but said he would closely examine the law, which will only become effective once he signs it.
"I shall look at the bill once it is forwarded to me with a view to identifying and addressing possible grey areas to ensure the new media law conforms to the constitution," a statement from the presidency said, quoting Kenyatta at a public rally near the capital Nairobi.
This draconian bill realises the media's worst nightmare, where government makes itself both judge and jury of what journalists say and how they say it.
Committee to Protect Journalists
Former Prime Minister Raila Odinga, Kenyatta's defeated rival for the presidency, urged him to throw out the new law.
"We can only ask him to have a change of heart and refuse to sign it because it is bad for the country and goes against the constitution," Odinga said in a statement.
If passed into law the bill would lead to huge fines against journalists and media organisations that violated a code of conduct.
MPs voted in a late-night sitting on Thursday to set up a new Communications and Multimedia Appeals Tribunal with the powers to impose penalties of up to $234,000 on offenders and even bar journalists from working.
The bill would also herald strict controls on radio and television broadcasts, with stations obliged to ensure that 45 percent of programme content and advertising is locally-made.
The passing of the bill comes as Kenya takes a string of measures to reinforce national security in the wake of the September attack by gunmen on the Westgate shopping mall.
Kenyan media drew the ire of authorities by broadcasting security camera footage of troops who were dispatched to the scene of the attack purportedly looting the upmarket mall.
Cyrus Kamau, managing director for Capital Group - home to CapitalFM, one of Kenya's most respected independent radio stations and news websites - called the law "draconian and very punitive".
The Daily Nation newspaper splashed its Saturday front page with the bold headline "No, Mr President!".
The US-based Committee to Protect Journalists (CPJ) called on Kenyatta to block the bill entirely.
"This draconian bill realises the media's worst nightmare, where government makes itself both judge and jury of what journalists say and how they say it," said CPJ East Africa representative Tom Rhodes.
"President Uhuru Kenyatta must veto the bill and safeguard Kenya's vibrant press," he added.