The tweets come after Trump was criticized over the weekend for doing little to address the two separate shooting massacres that left 29 people dead.
"Republicans and Democrats must come together and get strong background checks, perhaps marrying this legislation with desperately needed immigration reform", he continued.
Trump also said the nation must reform mental health laws to better identify "mentally disturbed individuals".
Adhering to what has become a GOP talking point of blaming mental illness and not firearms for mass shootings, Trump signaled he would oppose large-scale gun control efforts pushed by Democrats, saying, "hatred pulls the trigger, not the gun".
Trump responded to the attacks on early August 5 in an address at the White House where he denounced racism, bigotry, and white supremacy, and partly directed blame toward violent video games, saying that the "glorification of violence" must stop.
Earlier on Monday, he had called for stronger background checks on gun buyers.
Police said the suspected shooter has been charged with murder offences that can carry the death penalty, and a federal official said investigators are treating the El Paso shooting as a case of domestic terrorism. Just a few hours later, another gunman in Dayton, Ohio, killed nine people.
Whatever the political back-and-forth, or the re-energized presence of gun control talk on the presidential campaign trail, the very real consequences of gun violence were still being bared by victims badly injured in the two states.
Several other Democrats vying to face Trump in the November 2020 presidential election likewise blamed him for the attack in Texas, citing his rhetoric on immigrants.
Betts was gunned down within 30 seconds of the start of his rampage. President Obama had 32 mass shootings during his reign.
In a racist screed allegedly written by the suspected El Paso gunman, he expressed support for the Christchurch, New Zealand shooter, and denounced the increasing Hispanic population in Texas.
She said she wanted to tell the Republican president: "I need you to acknowledge that you've dehumanized people who are good and equal to all of us".
"Regardless of what happened here on Saturday, he has constantly demonized, vilified the type of people who live in this community", the El Paso county commissioner David Stout told ABC News on Tuesday.
Obama himself, who fought unsuccessfully for gun restrictions while in office, did not mention Trump by name when he urged Americans to reject divisive rhetoric.
The El Paso killings prompted 8chan founder Frederick Brennan, in an interview with the New York Times, to call for the site to be shut down.
The Del Sol Medical Center in El Paso confirmed the deaths during a news conference.
In Texas, where 22 were killed, authorities said the accused shooter hailed from a Dallas suburb a 10-hour drive away.