If you are using SQL Server 2016 or above, you can use sp_execute_external_script along with R. It has functions for Regular Expression searches, such as grep and grepl.
Here's an example for email addresses. I'll query some "people" via the SQL Server database engine, pass the data for those people to R, let R decide which people have invalid email addresses, and have R pass back that subset of people to SQL Server. The "people" are from the [Application].[People] table in the [WideWorldImporters] sample database. They get passed to the R engine as a dataframe named InputDataSet. R uses the grepl function with the "not" operator (exclamation point!) to find which people have email addresses that don't match the RegEx string search pattern.
EXEC sp_execute_external_script
@language = N'R',
@script = N' RegexWithR <- InputDataSet;
OutputDataSet <- RegexWithR[!grepl("([_a-z0-9-]+(\\.[_a-z0-9-]+)*@[a-z0-9-]+(\\.[a-z0-9-]+)*(\\.[a-z]{2,4}))", RegexWithR$EmailAddress), ];',
@input_data_1 = N'SELECT PersonID, FullName, EmailAddress FROM Application.People'
WITH RESULT SETS (([PersonID] INT, [FullName] NVARCHAR(50), [EmailAddress] NVARCHAR(256)))
Note that the appropriate features must be installed on the SQL Server host. For SQL Server 2016, it is called "SQL Server R Services". For SQL Server 2017, it was renamed to "SQL Server Machine Learning Services".
Closing Thoughts
Microsoft's implementation of SQL (T-SQL) doesn't have native support for RegEx. This proposed solution may not be any more desirable to the OP than the use of a CLR stored procedure. But it does offer an additional way to approach the problem.