Q: Can a employer decline a job offer if they run a background check and see an arrest but no charge

A: Using a job applicant's criminal background as a method for screening job candidates is problematic unless there is a specific law or legal regulation that disqualifies a person based upon specific criminal conduct. In many instances, prophylactic use of arrest and conviction records can be a pretext for discrimination based on race, color, national origin, and/or ethnicity. See EEOC guidelines: https://www.eeoc.gov/laws/guidance/enforcement-guidance-consideration-arrest-and-conviction-records-employment-decisions

Empirically, blacks and Hispanics in the US are more likely to be arrested and more likely to be convicted of crimes than either whites or Asians. Disqualifying job candidates based on broad facially neutral criterion (e.g. anyone arrested for a crime within the last five year, or anyone convicted of a felony) can nevertheless have a disparate impact on applicants based on race, color, national origin, and/or ethnicity.

Employers can drill down and make decisions based on the specific nature of an arrest or conviction as compared to the essential job functions of a position. For example, an employer can disqualify a job applicant for a position as bookkeeper based upon a previous arrest or conviction for embezzlement. Similarly, an employer might disqualify an applicant seeking a door-to-door sales position if the applicant was previously arrested for sexual assault.