Trump didn't propose a Muslim registry, it was proposed by a reporter during an interview, while Trump was talking about the wall.
Trump did call for a database for Syrian refugees coming to the US.
Trump's answers on the issue have been confusing, as it seems that he was always talking about the Syrian refugees registry. Trump never accepted the all Muslim registry, nor did he unequivocally deny it.
Politifact has an article that summarizes the issue and presents it step by step as it unfolded. A full answer would be a copy-paste of the entire article; to avoid that, I've put forth just a few excerpts here, but I highly recommend reading the article fully for a full view of the issue:
But Trump has said he didn’t propose such an idea -- a reporter did, and Trump just didn’t understand the question.
It all started on Thursday, Nov. 19, when a Yahoo News reporter asked Trump about his position on increased surveillance of American Muslims.
"France declared this state of emergency where they closed the borders and they established some degree of warrantless searches. I know how you feel about the borders, but do you think there is some kind of state of emergency here, and do we need warrantless searches of Muslims?" the reporter asked.
"We’re going to have to do certain things that were frankly unthinkable a year ago," Trump said.
The Yahoo reporter then asked Trump, "Do you think we might need to register Muslims in some type of database, or note their religion on their ID?"
Trump responded, "We’re going to have to look at a lot of things very closely. We’re going to have to look at the mosques. We’re going to have to look very, very carefully."
Here, Trump didn’t reject the idea of a Muslim registry, but he also didn’t give an affirmative "yes" that he wanted to create such a database.
The next day, an MSNBC reporter asked Trump, "Should there be a database or system that tracks Muslims in this country?"
"There should be a lot of systems," Trump responded. "Beyond databases. I mean, we should have a lot of systems."
Trump then digressed to talk about a wall along the southern border, before the reporter interjected, "But that’s something your White House would like to implement."
"I would certainly implement that. Absolutely," Trump said.
Here, we’re not clear if Trump is talking about implementing a wall or implementing a database.
While many headlines came out after this exchange saying Trump would "absolutely" require Muslims to register in a database, it’s not entirely clear that’s what he said. Trump was talking about building a wall along the border when the reporter asked if he would implement an unspecified policy -- "that" -- as president.
Through the end of the conversation, it’s possible Trump thought the exchange was about illegal immigration.
Later that day, Trump wrote on Twitter, "I didn't suggest a database -- a reporter did. We must defeat Islamic terrorism & have surveillance, including a watch list, to protect America."
After Trump’s tweet, Fox News asked him about his position on a Muslim registry.
"Let's hear it directly from you," said host Kimberly Guilfoyle. "Would President Donald Trump support a full Muslim database?"
"Basically the suggestion was made and (it’s) certainly something we should start thinking about," Trump said, repeating that the reporter presented the idea. "But what I want is a watch list. I want surveillance programs. Obviously, there are a lot of problems. … But, certainly, I would want to have a database for the refugees, for the Syrian refugees that are coming in because nobody knows where they're coming from."
Guilfoyle followed up: "So to be clear, you are not saying anything with respect to a religious database. You are talking about the Syrian refugees in light of the national security development affecting this country as we speak here tonight."
Trump said he didn’t hear the MSNBC reporter’s question clearly, "but even if I did, I mean, I want databases for the Syrian refugees that Obama is going to let in if they come in."
On ABC News’ This Week, host George Stephanopoulos asked Trump, "You did stir up a controversy with those comments over the database. Let's try to clear that up. Are you unequivocally now ruling out a database on all Muslims?"
"No, not at all," Trump responded. "I want a database for the refugees that -- if they come into the country. We have no idea who these people are. When the Syrian refugees are going to start pouring into this country, we don't know if they're ISIS, we don't know if it's a Trojan horse. And I definitely want a database and other checks and balances. We want to go with watchlists. We want to go with databases. And we have no choice."
Trump’s exchange with Stephanopoulos seems to be the clearest explanation of his position. No, he would not rule out a database on all Muslims. But for now, he wants a database for refugees.