First off, Republicans don't actually blame the government deficit on SNAP (food stamps.)
That aside, no, the representation of U.S. government expenditures in the pie chart in question is not even remotely accurate. As DavePhD accurately pointed out in his answer, the chart only shows what is termed "discretionary spending" in the federal budget, which doesn't even include food stamps (or almost any of the other social entitlement programs spending.)
Here is what federal expenditures actually look like:
Federal Expenditures for FY2015. Source: National Priorities Project, which sources from OMB (the White House Office of Management and Budget.)
The image in the question was also reviewed by Politifact and, due to its presentation of discretionary spending as if it were the total of federal spending, was rated false.
Additionally, the question asks about "U.S. government spending." When discussing government spending in the U.S., it's important to remember that federal spending is itself only a piece of the pie. The chart above and the chart in the question show only federal spending. Nearly all military spending is federal, however, state and local governments account for a large percentage of government spending in the U.S. in other sectors. Education, for example, is typically funded primarily by state and local governments. The same is true for road construction and maintenance.
The chart above shows the $3.7 trillion dollars of federal expenditures in fiscal year 2015. However, total federal + state + local government spending in the U.S. for that fiscal year is estimated to be about $6.4 trillion, according to usgovernmentspending.com. Official data from the government for these figures is not yet available, but they're in line with what would be expected based on prior year data, which is available from the U.S. Census Bureau.
When all U.S. government spending is included (state + federal + local,) education and transportation will be much larger shares of spending than when only federal spending is considered. Conversely, the share of military spending will only be about 60% of its share of federal expenditures, since nearly all military spending is from the federal government.
While Fiscal Year 2015 data is not yet available, here is a breakdown of the major spending categories for state and local government spending from the Fiscal Year 2014 data listed by the Census Bureau (XLS):
U.S. State and Local Government Spending, FY 2014, Own Work
Data Source: U.S. Census Bureau (XLS)
Overall U.S. Governmental Spending, Fiscal Year 2014
During FY 2014, state and local governments received $602 billion from the federal government in various grants, according to the Census link cited above. Unfortunately, the Census data does not break this down by category, so constructing a pie chart showing a breakdown for both federal and state and local spending by category isn't possible from this data. However, the following chart shows how federal spending categories looked as a percentage of overall governmental spending in the U.S. for Fiscal Year 2014:
Total U.S. Governmental Spending, FY 2014, Own Work
Data Sources: OMB for federal data (XLS) and U.S. Census Bureau for state and local (XLS)
The "State and Local Spending Minus Federal Grants" portion of the pie chart shows all spending from state and local governments minus the $602 billion of that which was funded by grants from the federal government. This subtraction is necessary in order to construct an accurate pie chart, since that $602 billion would otherwise be counted twice.