What does it mean when Nancy Pelosi criticizes the revised Republican Health-care plan as "doo-doo stuck to their shoe”?

Washington Examiner (April 27) carries an article titled “Pelosi: GOP will have 'doo-doo' on its shoes if Obamacare repeal," in which she argues;

“I think President Trump is really making fools of the members of Congress, of his own party. He's asking them to vote for a bill that is wildly unpopular in the country, is the wrong thing to do first and foremost. It's going to be doo-doo stuck to their shoe for a long time to come with terrible consequences for the American questions."

https://www.washingtonexaminer.com/pelosi-gop-will-have-doo-doo-on-its-shoes-if-obamacare-repeal-passes/article/2621457"

Though I cannot find “doo-doo” in Oxford Advanced English Leaners Dictionary at hand, Kenkyusha’s English Japanese Dictionary gives definition of “doo” as (slang) pee, crap.

Does “doo-doo stuck to one’s shoe” mean an indelible and foul stain? Is it a popular phrasing? What are easier and more decent alternatives to “doo-doo stuck to one’s shoe”?

"Doo-doo stuck on one's shoe" is a metaphor. If you are out walking and happen to step on a piece of the path that a dog has decided to foul you will find yourself with 'doo-doo' stuck on your shoe.

'doo-doo' is not easy to clean off your shoe. So you may find you are stuck with the foul stench and unhygienic extra material on the sole of your shoe until you are able to properly take your shoes off and wash them thoroughly.

That's the point the author is making with this base metaphor.

If congress vote for a bill that is wildly unpopular in the country, the fact they voted for it will be attached to them. It will affect their reputation and perhaps even conscience in much the same way 'doo doo' contaminates the environment of someone if they have happened to get it stuck to the soles of their shoes.

Doo-doo in this context means shit, no more and no less.

A person who steps in shit and can't get it off their shoe are likely to continue to find themselves in embarrassing situations - on the bus, on the elevator, pretty much anywhere where there are people around who'd be likely to be looking at the owner of the shit-stained shoe strangely.

Doo-doo is what kids (and their parents) call shit, imagining it's "less offensive" than the actual word. It is a euphemism.

Other euphemisms include poo, poo-poo, poop, ca-ca, etc.

The euphemism for a shithead is an alternatively intelligent person.

There is also a bunch of semi-euphemisms, including "feces" and "excrement."

I am going to answer only the last part of the OP's question, because the other answers did not address:

What are easier and more decent alternatives to “doo-doo stuck to one’s shoe”?

Let's put politics firmly aside. I venture to say that most people will not find the phrase indecent. Earthy, yes, but not indecent. One of the definitions of earthy in Dictionary.com is "coarse or unrefined".

As for an alternative, Pelosi could have said:

they will wear a scarlet letter on their suit jackets for a long time

but then very few would have understood her, and many of those think Hester Prynne was admirable.

In the context of strong partisan politics, doo-doo on their shoes hits the right note.

Literally, the phrase means that someone will have stepped in a pile of excrement.

In a country where a significant portion of people have driven by a farm (in a car), visited a farm, worked on a farm, or lived on a farm (small or large), the phrase evokes the feeling of disgust and dismay for everyone involved. First, the person who has stepped in the excrement has the visceral reaction to the accident of smooshing their foot down into the pile, leaving a clear mark of the sole of their shoe in the now-flattened pile. Second, everyone who comes into contact with the person (before they can change clothing and bathe) will be confronted with the aggressive, lingering smell of the mistake.

People who have strictly lived in cities may also have specific sense memory of this situation since cities use horses on roads for parades (on occasion) and horse-drawn carriages. Horses will defecate on the roads in these processes. Certainly, most native speakers are familiar with the phrase "horseshit", a particular brand of "shit" (which is a crass but generally acceptable euphemism for excrement, especially in the moment of accidentally stepping in it). The wheels of the aforementioned carriages may also go through such piles on accident.

However, the phrase in this particular evocation of that sense memory is awkward in my opinion. A possible explanation for this is that certain words on television in the United States of America are prohibited (at certain specified times; see the Federal Communications Commission's statement here: Obscene, Indecent and Profane Broadcasts) and carry with them a fine to the broadcaster. If less of them are prohibited now, that does not mean that they were not prohibited historically. Furthermore, this is how someone might talk to a child. The idea may be that saying "shit" around a child may encourage that child to begin using it frequently (since it is likely to get a reaction from people hearing it out of the mouth of a child).

The phrase "doo-doo" is also a replacement for "shit". To a native speaker, replacements such as this for obscene words and phrases ("swear words", "curse words", "cussing", etc.) may sound awkward. The level of awkwardness may be in the range of "at least a little ridiculous" to "absolutely ridiculous". There are some examples of this phenomenon here: Cursing Without Cursing. The part at the end of that video makes a callback to the somewhat nonsensical reason for why this even exists in the first place, but, since these are movie clips, they're fictional (and may be meant to parody the situation the language finds itself in).