They came from all corners of the US so that new US President Donald Trump would hear them loudly, and they called out for respect and dignity.

Hundreds of thousands of women marched through the streets of Washington and other major US cities to lead a global protest against Trump on Saturday, a day after his inauguration as the 45th president.

Estimated at a half million by the Women's March organizers, the largely peaceful demonstrations expressed women's discontent at Trump's comments and stances on issues involving women, immigrants, race and healthcare.

In Washington, marchers from across the country, many with pink cat-eared knit hats in reference to Trump's boast in a 2005 video made public last year, filled the streets around the National Mall and the White House. They held customized signs and chanted slogans such as "We are the popular votes."

Celebrities such as singers Madonna and Alicia Keys and actresses Scarlett Johansson and Ashley Judd spoke at the Washington protest.

Madonna's comment in which she said that she "thought about blowing up the White House" has resulted in a possible Secret Service investigation, according to published reports.

On Sunday, Madonna said on Instagram: "I am not a violent person; I do not promote violence, and it's important people hear and understand my speech in its entirety rather than one phrase taken wildly out of context."

Elaine Enders-Lang from Warsaw, Indiana, said: "I am not protesting, I am demonstrating. I want Trump to see that we are a united country, and we believe in the rights for everyone.

"I do think women's rights in this country are in jeopardy," Enders said. "I want respect for women. I want women to make their own choices. I don't think men should make choices for them, their bodies. I think he needs to apologize for the words he has said, and the way he treated women and minorities."

Amy Mulvihill from Baltimore, Maryland, who held a sign saying "Equal rights for all", said she came to stand up for women's rights. She said there's a general fear that Trump will continue to favor the elite and take away what Barack Obama was able to give to average working Americans, such as healthcare.

Cambria Ungaro from Virginia said: "I'm protesting the views of Trump, specifically his thoughts on women, his thoughts on immigrants and his thoughts on people of color. His disrespect of almost everyone in this country, his disregard for our well-being, our feelings."

For his part, Trump tweeted twice about the marches on Sunday:

"Watched protests yesterday but was under the impression that we just had an election! Why didn't these people vote? Celebs hurt cause badly."

"Peaceful protests are a hallmark of our democracy. Even if I don't always agree, I recognize the rights of people to express their views."

Husbands, boyfriends and children also attended the Washington march.

Scott Denslow, who came with his wife from Lousiana, said: "I'm here for women, for equality, for Black Lives Matter, for LGBT rights and for togetherness and unity."

Thousands also marched in New York, Chicago, Seattle, Los Angeles, Boston and around the world.

Waves of cheers rang out on the streets at the New York march. Mayor Bill de Blasio said some 400,000 took part.

Martha Bird, 51, traveled from New Jersey with her sign: bold, capital letters of the word RESPECT, with arrows pointing to every direction.

"The last several months demonstrated that there is an imbalance in respect. I'm here to show respect to my friends, to my family, and to people I know and I don't know," she said. "There are a lot of people who want to see positive change, just want to be constructive. Today is a very good example of that."

In Seattle, the number of marchers was estimated at 175,000.

Chen Weihua, Wang Linyan and Yuan Yuan in Washington, Hezi Jiang in New York and Linda Deng in Seattle contributed to this story.